Sources of the hadith
It is mentioned with the wording quoted in tafsir ibn Kathir in Sunan ibn Majah and similarly in Sunan ad-Darimi, in the Musnad of imam Ahamad 1 & 2, in at-Tabarni's al-Awsat 1 & 2 and in Musanaf ibn Abi Sahybah (Hadith #36):
‘O ‘Aishah, beware of (evil) deeds that are regarded as insignificant, for they have a pursuer from Allah. (i.e. accountability).”
Here some other (partly explanative) versions of the hadith:
“Beware of minor sins, like a people who camped in the bottom of a valley, and one man brought a stick, another man brought a stick, and so on, until they managed to bake their bread. There are some insignificant sins which, once (they accumulate) and a person is questioned about them, they lead to his doom.” (Musnad Ahmad and at-Tabarni in al-Awsat -in Arabic-)
Note that scholars consider keep on doing a minor sin equal to a major sin!
“Beware of minor sins, for they accumulate until they doom a man,” and the Messenger of Allaah (peace and blessings of Allaah be upon him) gave a likeness of that: “Like people who camped in the wilderness, and one man brought a stick, another man brought a stick, and so on, until they accumulated a lot, then they lit a fire and cooked their food.” (Musnad Ahmad and in at-Tabarni's al-Kabir and al-Awsat on the authority of 'Abdullah ibn Mas'ud)
Source of the last two translations: islamqa #47748
I might have missed some sources for example I couldn't make out a hadith like this in an-Nasa-i's compilation even if it was mentioned in many sources. But it is clearly a hadith.
Interpretation of the hadith
As Sindi in his commentary on Sunan ibn Majah has said (My own translation take it carefully):
قوله : ( ومحقرات الأعمال ) أي : ما لا يبالي المرء بها من الذنوب
(insignificant deeds) means all those sins a person doesn't care about (*)
(طالبا ) أي : مكلفا فعرض عليه أن يطلبها فيكتبها فهي عند الله تعالى عظيمة حيث خص لأجلها ملكا .
(a pursuer) means an accountable who is asked to write them down, so they are considered as big (sins) by Allah, as he specified an Angel for them.
(*) as one regards them as harmless.
Some scholars like al-Manawi further explained that small sins may lead to big sins. Al-Ghazali said something like the small sins pull each other until their owner is lead to a bad ending.
Imam al-Bukhari has quoted a similar hadith in his Sahih:
Anas said "You people do (bad) deeds (commit sins) which seem in your eyes as tiny (minute) than hair while we used to consider those (very deeds) during the life-time of the Prophet (ﷺ) as destructive sins."
In his fath al-Bary ibn Hajar quoted ibn Battal saying that the minor sins when increasing become major sins. He also quoted a narration on the authority of abu Ayub al-Ansari compiled by Asad ibn Musa أسد بن موسى in az-Zuhd الزهد saying that... (My own translation take it carefully):
إن الرجل ليعمل الحسنة فيثق بها وينسى المحقرات فيلقى الله وقد أحاطت به
a man does a good deed and has confidence in it and forgets about the insignificant (deeds) and when he meets Allah it is surrounding him
وإن الرجل ليعمل السيئة فلا يزال منها مشفقا حتى يلقى الله آمنا .
and a man is doing bad deeds and keeps worrying about them until he meets Allah in secure.
An example of a small sin and how it may lead to a major sin is quoted here:
Shaykh al-Islam Ibn Taymiyah said in Majmoo’ al-Fataawa (15/293):
Adultery is a major sin, but looking and touching are lamam (minor sins) which may be forgiven if one avoids major sin. But if a person persists in looking or touching, that becomes a major sin, and persisting in that may be worse than a small amount of major sin, for persisting in looking with desire, along with the connected feelings of mixing and touching, may be much worse than the evil of an isolated act of zina. Hence the fuqaha’ said concerning the witness of good character: he does not commit a major sin or persist in a minor sin… Indeed, looking and touching may lead a man to shirk as Allaah says (interpretation of the meaning):
“And of mankind are some who take (for worship) others besides Allaah as rivals (to Allaah). They love them as they love Allaah” (al-Baqarah 2:165)
The one who is in love becomes a slave to the one he loves. end quote. (Source: islamqa #47748
Other quotes from the same fatwa above:
Al-Ghazaali said: Repeatedly committing small sins has a great effect of blackening the heart. It is like the effect of water dripping onto rock, which will inevitably erode it, even though water is liquid and rock is solid. End quote.
Al-Nawawi (may Allaah have mercy on him) said in Sharh Muslim:
The scholars (may Allaah have mercy on them) said that persisting in a minor sin makes it a major sin. It was narrated from ‘Umar, Ibn ‘Abbaas and others (may Allaah be pleased with them) that there is no major sin with prayers for forgiveness and no minor sin if one persists in it.
What this means is that a major sin may be erased by praying for forgiveness, but a minor sin may become a major sin if one persists in it. End quote.
I hope I covered your inquiry well enough.
A helpful reference in Arabic fatwas islamweb #39543