EXP Points are effectively what a Pokémon gains after defeating an opponent (there are some exceptions to this rule, shown at the bottom of this page). This can cause a Pokemon to Level Up which then increases stats and will make them stronger. Gaining expierence is key to the game. Better stats mean you can defeat tougher Pokémon. Getting enough EXP Points also means you can evolve your Pokemon, if you meet the requirements to do so.
A list if of some items that will help with leveling:
Some items in the Pokémon Mystery Dungeon Series can also help you level up. Such as: