Hello, when a taxi driver says kaarp on it's own, is that just an abbreviated form of thanks? What is the literal?

thanks!

rather "yes", confirming, that he understood where you want to go

It's actually short for the extremely polite phrase 'ขอรับ', meaning 'may I be of service', but is usually used as ackowledgement of what the other person has said.

A detailed explanation can be found here.

He is acknowledging you. It can be pronounced 'long' or 'short'.