Hi Christian "to be keen on something" means you enjoy or you like an activity very much For example "Susana is very keen on shopping but her husband Ron is not so keen on it". To be keen on person means you like or love that person. Joe is very keen on Sandra, but I don't know if she is so keen on him". I hope this helps.

To be 'keen' on doing something means you really want to do it. Like so: Jennifer is keen on going swimming this afternoon. Meaning, she really wants to do that, she is eager to do it. http://dictionary.reference.com/browse/keen

I have heard it used to refer to being enamored of someone, as in "He was keen on her" but that usage seems antiquated. You wouldn't hear it nowadays.