It implies that someone is trying too hard to be cool, often by being too aware of trends and conforming excessively, making them seem unoriginal or even pretentious.
Think of it as someone who:
* Dresses in the latest, most expensive brands even if they don't actually fit their style.
* Knows all the latest slang and buzzwords, even if they don't quite understand them.
* Follows all the "rules" of what's considered cool, even if it makes them seem fake or unnatural.
In essence, "too school for cool" is a way of saying someone is trying too hard to fit in, and in the process, they end up coming across as unauthentic or even uncool.