When the Prophet Muhammad (PBUH) became prophet, he did not actively fight the Meccans until he immigrated to Medina after three years. Afterwards, he had multiple wars with the Meccans (e.g. the Battle of Badr, the battle of Uhud and the Battle of the Trench).
Why did the Prophet wait three years to battle them, while during this time the Meccans were hostile to the Prophet and the Muslims and killed many of them during these three years?
Allah never commanded the Muslims to fight before they migrated to Medina. Instead the Prophet (pbuh) was commanded many times to remind the people of the truth.
"So remind, if the reminder should benefit" - Quran 87:9
The longer Muhammad (pbuh) preached, the more people accepted Islam. Although the Muslims were being persecuted in Mecca, they still had enough protection for the message to be conveyed to the public. It was only after Abu Talib's death and an assassination attempt, when he (pbuh) and the rest of the Muslims had to end this kind of preaching and migrate.
If you look at the story of other Messengers such as Nuh (pbuh), or Musa (pbuh), then you will see the same methodology. First the messengers warn the disbelievers for a period of time (determined by Allah), and then the disbelievers (if they did not accept the message) face the consequences of their disbelief immediately afterward.
I am not really sure what you mean by 3 years. The Prophet ﷺ stayed in Makkah for around 13 years. Then he migrated to Medinah and military activities started soon after migration, the expedition of Hamza was dispatched in the first year - similarly there were several other military activities before the Battle of Badr. The rest of this answer assumes that you meant to ask as to why the Prophet ﷺ did not fight before the migration to Medinah.
The Prophet ﷺ did not fight the disbelievers during the early Makkan period of Islam because Allah had commanded him to be patient and to pardon, overlook and turn away from the disbelievers in verses such as 73:10, 41:34, 5:13, 16:125, 24:54, 25:63, 88:22, 3:186, 50:45, 45:14, 15:94, 15:85. Later He abrogated it and permitted and then obligated fighting.
This is evident from the Quran:
ألم تر إلى الذين قيل لهم كفوا أيديكم وأقيموا الصلاة وآتوا الزكاة فلما كتب عليهم القتال إذا فريق منهم يخشون الناس كخشية الله أو أشد خشية
Have you not seen those who were told, "Restrain your hands [from fighting] and establish prayer and give zakah"? But then when fighting was ordained for them, at once a party of them feared men as they fear Allah or with [even] greater fear.
And hadith:
فقال " إني أمرت بالعفو فلا تقاتلوا " . فلما حولنا الله إلى المدينة أمرنا بالقتال
He said: “I have been commanded to pardon, so do not fight.” Then, when Allah caused us to move to Al-Madinah, He commanded us to fight.
And consensus:
ولا خلاف في أن القتال كان محظورا قبل الهجرة
There is no disagreement on the fact that fighting was forbidden before the migration
— Tafsir al-Qurtubi also see Tafsir al-Jassas
شرع الجهاد بعد الهجرة النبوية إلى المدينة اتفاقا
Jihad was prescribed after the Prophet's migration to Medinah, by agreement
As for the wisdom behind it, then it could be that the Muslims did not have enough strength to effectively fight at that time. And it could also be a period of respite from Allah to allow the disbelievers time to be repent and reform.
Note: Quran references are incremented by one as bismillah is counted as the first verse.
[22:40] Permission to fight is given to those against whom war is made, because they have been wronged — and Allah indeed has power to help them — [22:41] Those who have been driven out from their homes unjustly only because they said, ‘Our Lord is Allah’ — And if Allah did not repel some men by means of others, there would surely have been pulled down cloisters and churches and synagogues and mosques, wherein the name of Allah is oft commemorated. And Allah will surely help one who helps Him. Allah is indeed Powerful, Mighty —
Holy Prophet (saw) never wanted to engage in war. He wanted even his enemies to see the truth and accept Islam. It was only when persecution was raised to the stage that not defending themselves was equivalent to suicide was when Islam was forced into wars of self-defense. According to the Holy Quran they were permitted because Allah decided to give them victory over their oppressors. Allah sought to prove that Muslims are those who submit in hardship while they are weak, but Allah indeed has the power to help them as he is the most powerful and mighty. It proves that Muslims were capable of being completely submitted to Allah regardless of conditions, such that they could take immense persecution and not retaliate purely for the sake of Allah. Once they were permitted it also showed they were not fighting out of cowardice, but only held back as Allah had not permitted them. There is a misconception that Allah had waited until Muslims were powerful enough, but this is nonfactual and utterly false. Muslims were in these times very weak and heavily outnumbered so much so that some of them even used wooden swords to defend themselves.
Here is a more detailed answer very very well explained of details and wisdom of Allah in putting Muslims through 13 years of such persecution, please listen: Why was the permission to go to war given to Muhammad long after the Muslims began to be persecuted and can wars be fought for non-religious reasons?
As you know about Battle of Bader,Uhud, Khandaq(first three wars ) after hijrah (migration to Madinah) ,all these wars was imposed by People of Makkah(non-Muslim). They were afraid from Islam and think that Islam will dominate their religion, so they attack on Madinah in order to finish Islam. Prophet MUHAMMAD (Peace Be Upon Him) and his companions only protect Madinah and their life.
In Makkah Prophet MUHAMMAD (Peace Be Upon Him) do peaceful teaching but when Prophet MUHAMMAD's(SAW) and his believers life become so bad then they all migrate to Madinah.
