My dog used to chase people on a bike a lot.

My dog used to chase people on a bike a lot.
It got so bad, finally I had to take his bike away.

The humor in this joke lies in the ambiguity of the setup. The setup could be understood as "My dog used to often chase people who were riding on a bike", and it could also be understood as "My dog used to often ride on a bike while chasing people." Since the first sense of the sentence is normal, and the other one is absurd the listener hears this sense first.

When the punch line occurs, the listener has to reevaluate the first sentence to make sense of it. Additionally, the other version of the sentence is pretty much absurd. Dogs can't ride bikes.

This joke could be told as if it were not a joke, because the first sentence is a fairly normal thing to say. If that were done, the surprise of the second sentence would probably make it funnier.