You can read about Abu Bakr's reason in @UmH's answer. You can read the Shia account at Wikishia which also provides an English version.
In this answer I offer a different insight as to the real reason for Fatima's protest beyond a narrow focus on only the juridical aspect of the question. Because what I see in the present answer and in other Sunni apologetics (and even arguments by some Shia) is a fixation on the dichotomy of whether Prophets leave inheritance or not and how that adds up with the Quran. What I don't see from Sunnis but I do more often from Shia is proper attention to the greater context of the dispute which raises questions that fails the Sunnis but can be fully made sense of in the light of the Shia opinion and sources.
The Sunni accounts here as with many other tendentious issues leave many questions. The narrations that we have don't seem balanced. We see Abu Bakr's and Umar's argument in much greater detail than the argument by Banu Hashim. For example what I read here in Sahih Muslim is that Abbas and Ali went to Umar’s court apparently only to listen to Umar pontificating them about the religion of the Prophet ﷺ and promptly answer his blunt yes/no questions without making any comments! So I tend to think this is not a tradition that can give us the whole picture but what it does tell us is how Ali and Abbas, according to Umar’s own narrated words, thought that Abu Bakr and Umar were “liar, treacherous and dishonest” in the issue of inheritance.
That for me betrays a very serious grievance on the part of a people who I know were far more generous, forgiving and pious to be so much aggravated by a mere worldly dispute over material inheritance of a prophet of Allah ﷺ. What I care less about is the subjective nuances, according to this or that companion/scholar, of the terms of the Prophet's inheritance that may give weight to this or that side's argument. Because there is a more fundamental and significant question here: how could the dignified Banu Hashim and illustrious characters like Ali (as) and Fatima (as) have been so outraged?
Now in the light of my overall picture of the religion of the Prophet ﷺ I can actually grant that, by observation of a reasonable moral in suggesting that prophets’ properties are charities, that a hadith attested to only by Abu Bakr and his sworn comrade Umar may have merit. All Prophets after all spent from their wealth in charity as much as they could as they were not attached to their wealth and didn’t think big about being rich and pandering their children. That profane mentality would run contrary to their spiritual message and character.
But the question that still troubles me is: can we make a statement, intended to make a moral point, absolute? Really can you commit that Prophets don’t leave behind any inheritance at all given that they are humans after all and liable to basic facts of life?
Because then I’ll ask you who inherited Prophet’s house? Well, his wives and family! So is the hadith contradicted now? No, the moral still stands, but yes, if you interpret the hadith in absolutes and disregard its intended moral!
What else do the Prophet ﷺ, indeed all Prophets, left behind as inheritance? His caliphate? His religion? His Sunnah? His conduct?
We know for example that Prophet Sulaiman ﷺ inherited both prophecy and rule from his father, Prophet Dawoud ﷺ and certainly Dawoud’s wealth and properties. That’s just natural. Familial inheritance has been a Divine law since Adam and common in the secular world and and there’s no reason why Prophet Muhammad ﷺ be an exception. In case of the prophets though a big part of the inherited wealth is spent on the cause of religion and charity not exclusively for personal use. But that's different from saying they won't leave a penny for their daughter and family and that his family should sustain on the mercy of Abu Bakr. There was a reason Prophet didn't want his distinguished family to live on charity from the people. I don't see how charity from an unapproved government is any more dignified.
Now with the proper conceptual background set, let’s look at the actual dispute here in the light of its immediate background because that’s how you should interpret yet another quarrel between the caliph and what seems like the eternally aggrieved Banu Hashim!
Abu Bakr who has just imposed himself as caliph on Muslims despite facing strong resistance by the Ansar, Banu Hashim and many Muslims outside Media (who were lumped together with the rebelling apostates and repressed wholesale) is challenged again upon the immediate decision that he makes after securing the caliphate. The challenge comes from the same group: Banu Hashim who think that not only Abu Bakr has imposed himself as their ruler on them without consulting them but now he is taking away what they say was their inheritance.
So do you see the point here? Banu Hashim think that they are being increasingly dispossessed of their status and position that they rightly had when the Holy Prophet ﷺ was alive. They were not consulted on the caliphate and now suddenly the Prophet’s properties that would have gone to them with the caliphate had they succeeded it, is also now confiscated by the caliph!
That’s really the crux of the matter. Arguing over the terms of inheritance is pointless because anyhow the details depend on the person in power and his deliberations and "ijtihad". So it is one thing to argue over how Prophet's inheritance must be ideally spent but another to exploit a misplaced hadith to justify taking control of properties that had been just at the mercy of the Prophet ﷺ and his family!
That’s why to me Abu Bakr's and especially Umar’s lecturing Banu Hashim and setting conditions on expenditure of the Prophet’s possession by them come, with all due respect to our Sunni brothers, as quite arrogant and belittling to be frank! For, for God’s sake! Banu Hashim in general were never known for being materialistic or selfish or ignorant of the religion. To the opposite, their famed generosity goes back to the days of Jahiliyya when Abu Bakr and Umar were nobodies! As for Ali and Fatima in particular nobody even gets near to them in terms of knowledge, selflessness, piety, asceticism and good-will and yet they are being pontificated and patronized on the question of inheritance by a man that was infamous for his often harsh and reckless attitude in dealing with the people, forgetting about the reports of his even more radical actions against the Ahl al-Bayt!
Now we get to the better part of the answer. Shia who have been dedicated to collecting and keeping the record of the statements and stories of the purified Ahl al-Kisa and their descendants make it clear that Fatima (as)’s claim to Fadak was not an isolated financial dispute. It was part of what she saw as the jump start of a great deviation from the religion of the Prophet ﷺ.
In a lengthy moving superbly eloquent sermon, known as Kutbat ul-Fadakiyya, that the majestic daughter of the Holy Prophet ﷺ gave in the mosque in protest, after praise of the Lord and recount of the Prophethood, she is raising alarm on the real nature of the tragedy that has befallen the Ummah. Dispossession of Ahl al-Bayt from rule, religious authority and whatever else that came with them (Fadark, inheritance, etc), is seen as a great treachery against religion of Muahmmad which was established by his sacrifices and those of his family members against the often wavering and unfaithful Muslims. She is warning that after Arabs were saved from their abject situation during the Jahiliyya, the signs of hypocrisy and unfaithfulness are rising again after the Prophet’s death:
You were despised outcasts always in fear of abduction from those around you. Yet, Allah rescued you through my father, Muhammad after much ado, and after he was confronted by mighty men, the Arab beasts, and the demons of the people of the Book who, whenever they ignited the fire of war, Allah extinguished it; and whenever the thorn of the devil appeared, or a mouth of the polytheists opened wide in defiance, he would strike its discords with his brother Ali, who comes not back until he treads its wing with the sole of his feet, and extinguishes its flames with his sword. Ali is diligent in Allah's affair, near to the Messenger of Allah, a master among Allah's worshippers, setting to work briskly, sincere in his advice, earnest and exerting himself (in service to Islam); while you were calm, gay, and feeling safe in your comfortable lives, waiting for us to meet disasters, awaiting the spread of news, you fell back during every battle, and took to your heels at times of fighting.
Yet, when Allah chose His Prophet from the dwell of His prophets, and the abode of His sincere (servants); the thorns of hypocrisy appeared on you, the garment of faith became worn out, the misguided ignorant spoke out, the sluggish ignorant came to the front and brayed. The he camel of the vain wiggled his tail in your courtyards and the your courtyards and the Devil stuck his head from its place of hiding and called upon you, he found you responsive to his invitation, and observing his deceits.
He then aroused you and found you quick (to answer him), and invited you to wrath, therefore; you branded other than your camels and proceeded to other than your drinking places. Then while the era of the Prophet was still near, the gash was still wide, the scar had not yet healed, and the Messenger was not yet buried, [you made a} a hasty undertaking as you claimed, aimed at preventing discord, [but] surely, they have fallen into discord! And indeed Hell surrounds the unbelievers.
How impossible! How could you do this? For Allah's Book is still amongst you, its affairs are apparent; its rules are manifest; its signs are dazzling; its restrictions are visible, and its commands are evident. Yet, indeed you have casted it behind your backs! What! Do you detest it? Or according to something else you wish to rule? Evil would be the exchange for the wrongdoers! And if anyone desires a religion other than Islam (submission to Allah), it never will it be accepted from him; and in the hereafter, he will be in the ranks of those who have lost. Surely you have not waited until its stampede seized, and it became obedient. You then started arousing its flames, instigating its coal, complying with the call of the misled devil, quenching the light of the manifest religion, and extinguished the light of the sincere Prophet. You concealed sips on froth and proceeded towards his kin and children in swamps and forests, but we are patient with you as if we are being notched with knives and stung by spearheads in our abdomens, yet now you claim that there is not inheritance for us! What! "Do they then seek after a judgment of ignorance? But how, for a people whose faith is assured, can give better judgment than Allah? Don't you know? Yes, indeed it is obvious to you that I am his daughter.
O Muslims! Will my inheritance be usurped? O son of Abu Quhafa! Where is it in the Book of Allah that you inherit your father and I do not inherit mine? Surely you have come up with an unprecedented thing. Do you intentionally abandon the Book of Allah and cast it behind your back? Do you not read where it says: And Sulaiman inherited Dawood'?
And when it narrates the story of Zakariya and says: "So give me an heir as from thyself (One that) will inherit me, and inherit the posterity of Yaqoob" And: "But kindred by hood have prior rights against each other in the Book of Allah"
And: Allah (thus) directs you as regards your children's (inheritance) to the male, a portion equal to that of two females' And, if he leaves any goods, that he make a bequest to parents and next of kin, according to reasonable usage; this is due from the pious ones.' You claim that I have no share! And that I do not inherit my father! What! Did Allah reveal a verse regarding you, from which He excluded my father? Or do you say: `These are the people of two faiths, they do not inherit each other?!' Are we not, me and my father, a people adhering to one faith? Or is it that you have more knowledge about the specifications and generalizations of the Quran than my father and my cousin Imam Ali? So, here you are! Take it! Its nose rope and saddled! **But if shall encounter you on the Day of Gathering; (thus) what a wonderful judge is Allah, a claimant is Muhammad, and a day is the Day of Rising. At the time of the Hour shall the wrongdoers lose; and it shall not benefit you to regret (your actions) then! For every Message, there is a time limit; and soon shall ye know who will be inflicted with torture that will humiliate him, and who will be confronted by an everlasting punishment. (English translation of the sermon)
This was already very intense and devastating of a complaint, but if you read it in full you learn why Fatima (as) spent the rest of her life in depression until taking her grievance to the Messenger of Allah (as). She was already deeply hurt by the loss of the prophet of mercy but also had to endure the cruelties she faced after on and the dark vision for the Ummah that her father had nourished with tears and blood.
This sermon has over twenty isnads in various sources such as Balaghatu n-Nisa of Ibn Taifur, a third century Sunni man of letters and Kashfu l-Ghumma of the Shia scholar Ali ibn Isa Arbili.
