Essentially it replaces EV training. Completing the different trainings reward points in that specific stat. So if you did the speed training, you'd get speed points. Then when your pokemon levels, it will gain more points in the stat you've leveled. It used to be that you had to go out and fight specific pokemon to EV train. Each pokemon gives specific EVs. So in Black and White, if you wanted to get speed points, the easiest option was to use surf and fight a lot of basculin. But that was incredibly boring, took a long time, and required you to keep a tally (since the game didn't show you your accumulated EVs). With super training, you can do it all as soon as you get a new pokemon, and do it without having to battle.
For those who have never EV trained, there used to be an image that explained it well, but I can't find it, so gonna use words. Imagine you had two of the exact same type of pokemon -- let's say a Charizard. Both were the same level, both were the same nature, but one had been EV trained (or in this case super trained) and the other hadn't (how you got a charizard without any EVs is beyond me, but let's ignore that). If you compared the two, the one that had been EV trained would have higher stats at that level. As you play the game, beating pokemon and leveling your pokemon up results in accumilating EVs. The difference with EV training and now super training, you can pick which EVs your pokemon accumulates so it can excel in certain traits. So take Delphox. Delphox is clearly designed to be a special attacker. Once it evolves into Delphox, I don't think it gains any physical moves (once it evolves into Braixen I think, actually..). So if you had a Fennekin, you would want to super train its special attack so that you'd be boosting that stats that are most relevant to it.