"keep them keen" comes from a controversial expression : "Treat 'em mean, keep 'em keen". "em" is short for "them". It is said in the context of romance. The idea is that if a man treats women unkindly (or vice versa), he will actually become more desirable, rather than the reverse. No doubt the expression was used here with a lot of bravado.

"Get out of here" is kind of a slang phrase for saying "You're joking!" or "Don't be stupid!". It's a common phrase, so it's used a lot in books. As for "Keep them keen" I've actually never heard anyone say that before, but in the context of the sentence, I can guess that it means "I don't like to be with them often, I just hang around with them casually." or something like that.