Of a fictional character, to know the tropes of the type of story in which one appears, and to be able to use that knowledge effectively.
Savvy is, of course, understanding. If you imagine Johnny Depp as Captain Jack Sparrow when you hear the word, you've got it. Savvy?
Defining genre is a game at which even literary giants struggle, especially in the battle to draw or erase lines between 'literary' fiction and the purportedly second-class genre fiction. For our purpose here, though, genre is basically anything that lives in a named subsection of your local chain bookstore: Science Fiction, Fantasy, Mystery, Romance, and so on. Each of these has certain well-known conventions of which the reader is well aware.
If a character in the story exhibits awareness of the conventions and uses that awareness to guide their actions, or at least to avoid mistakes, they are genre savvy. This does not necessarily imply breaking the fourth wall, Deadpool-style, but talking to the audience about the plot of the story is definitely a form of this. For example, if you know enough not to split up your group in the face of danger, not to walk backwards into a darkened room, and not to go into the house that looks like a skull, you have horror genre savvy*.
It goes without saying that a quick shag in the woods is right out.
*Or you have both kids and Lego in the house.
References: I consulted TV Tropes, of course, which goes on and on and on about this. Go there for the thorough treatment, stay for the fun links, get lost for hours. ANOTHER VISITOR! STAY AWHILE! STAY...FOREVER!