The Japanese have this thing about collecting things and then battling them (see Digimon, Bakugan, et al.) The game was originally inspired by insect collections. I don't care much for the in-universe explanations, I just think it's fun.
Remember that it's an RPG. You fight things to get stuff and grow stronger, to get better stuff and fight bigger stuff. The fun part of it is the whole rock-paper-scissors strategy that's been hugely expanded; for example, Water beats Fire, Fire beats Grass, and Grass beats Water, but there's 14 other types too.
Pokemon can have one or two types (so Bulbasaur is Grass/Poison.) Attacks that have the same type as the Pokemon are stronger, but they can learn all types of attacks, and they only can know four moves at a time. So, if your opponent sends out a Fire-type, you might send out a Water-type to counter, only for him to use an Electric-type attack and hit you first.
Add in your standard status effects like Sleep and Poisoning, infinite randomly generated encounters, a multilayered Pokemon and move set management system, and an entire metagame based on breeding, and you have an incredibly deep and strategic game for both single-player advancement and multi-player competition.