What I have seen is a lot of kids really take off (be successful) because they have one-on-one attention, and they are up to grade level, if not beyond (higher than that), as a result of having been in the program (in a study schedule that helps them to progress). As I understand it, the statement means that the children become successful because they receive one-on-one support while they are in a special program. As a result, they reach a certain level or higher (if not beyond).

It means the children that have received one on one training, their level is at their expected grade or better. 'if not beyond' means more/higher/better. For example, "He had around 10 cats, if not beyond" means he had 10 cats or more (11,12,13... Cats).