A higher youth academy can accomodate and nurture more youngsters, so that your club can get more players coming through from the youth ranks. More youth players will be produced by a higher youth academy in a given time span. Also a higher youth academy will develop young talents faster and they will be ready for professional football at a much younger age.
Once a youth player in your academy is ready, he will acquire a license to play professional football. You will get a notification that the player is ready, and you can have a look at him to decide whether your club needs his services, and to promote him to your first team squad in that case.
You can promote one youth player per season for free, next one will cost 50% of the player's value, the next 100%, 150% and so on. The list of youth players waiting for your decision can be seen in your Facilities page, in the Youth Academy tab. Youth players do not stay in your academy forever, if you do not promote him for a long time, he will decide to leave the academy to find another club for himself.