Because they're fucking awesome and people tend to love em'

May as well ask why sugar and fat are so popular in food

You dig giant robots. I dig giant robots. We dig giant robots. Chicks dig giant robots.

Because mech has rigid body and gamedev likes controlling rigid body

I personally prefer big, clunky, stompy mechs (ala mechwarrior) over the super nimble ones (Armored Core, Gundam), but they're all great nonetheless.

Because human combatants means blood and gore and censorship or brutality.

Robots allow extreme violence, without the moral issues.

Because they increase the scale of combat to allow for michael bay like moments without having to write a character with nuclear weapon power levels