It means do a little bit more than compromise.
Say I took a woman out for a meal, and at the end of the meal I went to pay, she said, "Wait! Let me pay! You paid last time".
Being a gentleman I may say, "No, I insist that I pay".
She may reply, "Ok, but meet me half way, let me buy the drinks".
Well, ok, thank you.
THEN she pays for the taxi on the way home, she met me more than half way because if you see how much I spent on our night out, and how much she spent on the night out, she paid more than me.
:)