How do universities know how many/what AP courses you take?

from your high school transcript

If you don't know how educational institutions know your grades and classes you probably don't know what a transcript is===
It is an accounting database of your grades and every educational institute you have attended that can be accessed and sent to any institution you are desiring to attend for educational purposes

You are required to send in your high school transcripts. That will show all classes you took, at what level the class was at and you grade in each class.

they get sent from your high school.. or theykeep track.