Here’s the thing, there are actually a lot of reasons as to why we might want to colonize the asteroid belt. However, the main reason why we would want to colonize the asteroid belt is so we could mine it for precious materials.
That’s right, dissimilarly to the Earth, asteroids have a lot of valuable metals that are on the ground. When it comes to the Earth, it is true that these valuable materials will tend to be buried underground - asteroids are not like this. In addition to their mining properties, asteroids could also be used as a means of expanding our knowledge of space.