Have a look:
Consider a brick of a certain material X.
Looking cloosely, the molecules, forming your brick, vibrate; they are almost "fixed" in a place but vibrate around it.
When you increase the temperature of your brick you basically give energy to your molecules and, put it simply, you kick them so they vibrate with bigger amplitude "pushing" the surface of the brick. As a consequence they occupy a bigger volume and the brick as a whole expands!