For example, say you like someone (i.e. fancy them) and want them to ask you out on a date, but you have brown hair and you know that this person ONLY likes redheads (hypothetically), so when you're talking to your friend about the person you like, you might say: "Well, I know he wouldn't fancy me, he only dates redheads." Hopefully, you won't ever need to use this sentence :)

This term is an old fashioned way to say that someone was not interested in you. Nowadays if you are in that situation you say something vulgar or slang depending on where you live.

Imagine if you were speaking with a friend and your friend said you should ask a guy out on a date. If you do not think that the guy would like to go out on a date with you, you might reply by saying "he wouldn't fancy me". That simply means, "he wouldn't like me".