Learn Turkish: Biz o gün anaokulunda çok eğlenmişiz. - Apparently we had a lot of fun at kindergarten that day.

This is a very common use of the -miş past tense in Turkish! You use it for things you did but don't personally remember, like when you were a baby or a very young child. Here, the speaker is likely looking at an old photo or hearing a story about their kindergarten days, so they say eğlenmişiz ("apparently we had fun") instead of the direct past eğlendik ("we had fun").

How exactly is the word anaokulunda built?

It is a compound word with a location suffix attached. It joins ana (mother/main) and okul (school) to make anaokulu (kindergarten). To add the "at/in" suffix (-da), you first need the buffer letter -n- because the compound word ends in a possessive suffix (the -u). So it becomes anaokulu + n + da.

Does çok mean "very" or "a lot" in this sentence?

It means "a lot" here. While çok often means "very" before an adjective (like çok güzel - very beautiful), when it is used directly before a verb like eğlenmek (to have fun), it acts as an adverb meaning "a lot" or "much".

Since the verb ends in -iz, is the word biz at the beginning required?