Questions & Answers about Я записываю всё, что говорит учитель.
Записываю means I write down / I note down. It comes from записывать, which often means writing something down so you do not forget it.
That is different from пишу, which is the more general I write.
So:
- Я пишу письмо. = I am writing a letter.
- Я записываю адрес. = I am writing down the address.
In your sentence, записываю is used because the idea is noting down what the teacher says.
Записываю is the 1st person singular present-tense form of the imperfective verb записывать.
It matches Я = I, so the ending shows I am doing the action:
- я записываю = I write down / I am writing down
- ты записываешь = you write down
- он/она записывает = he/she writes down
Because it is imperfective, it can describe:
- an action happening now: I am writing down
- a repeated or habitual action: I write down