I'm better than what they have me doing.
Why do we speak like this? What is the role of "have" in the sentence? Why do we use ing after verb?
This is definition #6 at dictionary.com
to cause to, as by command or invitation: Have him come here at five.
They're ordering you to do something.
We add -ing to the action to make it a gerund, so we can treat it as a noun. This is necessary for the comparison in "I'm better than".
"have" has two meanings, possession or action.
"I have a car." You possess a car. This isn't an action, it is a statement of possession.
"I have taken a bath." You took the action of getting in the bath and bathing.
"they have me doing" utilized the second meaning of taking an action. They are making you take an action, and that action is.... doing something.