Why do Pokemon say their own names?

What a great question! There is some variance in the "calls" or sounds Pokemon make depending on whether you are reading the manga, watching the anime, or playing the games. The question as to why Pokemon say their names has not been answered definitively-- at least not in any English-language resources-- but there are a few theories we can consider. As we go over these theories, bear in mind that the Pokemon franchise has a lot of "nonsense" or silly components, like a ten-year old Ash being sent off into the world. The mystery of Pokemon calls might just be another piece of nonsense!

If we think "in-universe," it is possible that Pokemon were named for their distinctive cries. Perhaps the names for Pikachu or Squirtle did not exist until someone heard their calls. In our own world, some  animals are named in this way. For example, the Whip-poor-will bird is named for the distinctive sound it makes.

If we think "outside" of the universe of Pokemon and from the perspective of people who create and engage with media, having Pokemon say their own names could be a matter of ease. It would take up a lot of time to think of distinctive calls for all of the many Pokemon in addition to naming them and creating their attacks and abilities. It's possible the creators involved just wanted to keep it simple when it came to the sounds Pokemon make.

Alternately, it has often been suggested that Pokemon say their own names so that fans can more easily learn the names of different Pokemon. 

Not all Pokemon say their own names or are limited to this type of call. Charizard emits a large growling sound, and Meowth has the capability to speak in full human sentences.

Again, depending on the media we are referencing, some Pokemon don't say their names at all. In the anime, most Pokemon have the same name and call. In the manga, some Pokemon say their names, but not all. In the games, very few Pokemon are given a distinct call that isn't some kind of electronic noise-- Pikachu being an example of this. The Pokemon franchise began with the video games, so it is possible that when the anime and manga adaptations were created, artists wanted to extrapolate on the behavior of central characters like Pikachu.