To "take someone up on" something means to accept what they've offered. For example, if someone invited you to have dinner together, and you agreed, you can describe it this way:

I took him up on his offer to take me out to dinner.

This also works if someone offers to sell you some of their stuff:

He offered to sell me his car for two thousand bucks. You think I should take him up on it?

And you can take someone up on a bet that they've announced:

He bet me twenty bucks that I couldn't get a perfect score, and I took him up on it.