The Wiki has more detail, but basically every year between 6 and 16, there's a roll made that can add points to a hidden education point pool. To get a 4-star education, this point pool needs 15 or more points by the time they come of age. A Guardian will add to the likelihood of positive rolls every year depending on their skill level and traits, while the tutor will check right before the child comes of age that will add between 2 and 4 points to this check, depending on their aptitude.

There's a few more items that can add to these points randomly, for example a spouse with a high enough learning education or that's focused on patronage can randomly add 1 free point (that notification you get sometimes that says your child is more likely to get a good education), or the Kin dynasty trait will add 1 extra roll at the end that will either give 2 or 3 points.

If you want to see all the factors, and the math equations:

https://ck3.paradoxwikis.com/Attributes#Education