Leveling up a Pokemon in Day Care causes it to learn new moves, as well as gain stats just like normal. The only thing you can't do as oppose to leveling up in battle / via Rare Candy etc. is that it can't evolve. Also, leveling up by Rare Candies or Day Care doesn't make it gain any EVs, while by battling trainers or wild Pokemon does.
This not only applies to Pokemon Emerald, but also the other new as well as old games.
Source: Experience and Bulbapedia article