Bismillah Hir Rahmaanir Raheem

Beginning of Tafseer

The Prophet sallallahu `alayhi wa sallam was the first to receive Wahy and the first Mufassir.
During the time of the Sahaba, they did not ask many questions about the meaning of the Quraan for a number of reasons.
(a)  All the Aayats were relevant.
(b) All the incidents were pertinent.
(c)  They generally understood it.
Many of the Aayahs that were revealed, dealt with issues that directly or indirectly involved the Sahaba, so there wasn’t any need for much explanation.
Due to this, there were very few questions, so very little Tafseer from the Prophet sallallahu `alayhi wa sallam. However, with regards to delicate issues, he would explain it.
E.g. When Aiesha radhiallahu ‘anhaa asked about ‘Hisaabay Yaseera’ (84/8) .The Prophet sallallahu `alayhi wa sallam explained that the verse signified that, ‘The deeds will merely be presented before Allah without being questioned. As for the person whose deeds are scrutinized, he will never be able to escape torment.’ (Maariful Quraan vol8 pg 743)
The Sahaba would thereafter pass on what they heard from the Prophet sallallahu `alayhi wa sallam, to the Taabieen.
During the time of the Taabieen, there was no book of Tafseer, rather it was transmitted just like how Hadeeth would be narrated. Hence many books of Hadeeth would include a chapter on Tafseer.
It was only after the period of the Taabieen, that  Tafseer was compiled into a book form, with the Aayats of Quraan and their explanation together in one seperate book, similar to what is present today. (E.g. Ibn Katheer and Ma’aariful Quraan etc.)