The feeling is what got to me as a child. Starting as a boy, in a small city. You get your Pokémon, and go ahead, discovering the good and bad of the world, only you and your partners. You become stronger, with your Pokémon friends. You choose your team, from the wide variety of Pokémon at your disposal, and win with them by your side. And, just like that, you see te entirety of the world, you face it's hardships and become a master. That feeling of progression. The feeling that you get of "growing up", alongside your Pokémon. Overall, the feeling of the games is what makes them so magical to me. From a young boy with little understanding about the world and Pokémon to a Pokémon master, that journey is forever engraved in my memory.
While I was typing this, I realized that the reasons why I like Pokémon are pretty similar to the reasons of why I like Earthbound. Go figure...