"Awfully" is an adverb that means something in between "very" and "too". You use this when you're talking about something that's almost over the limit of being too much. In the example above, saying that the woman "seems awfully young" means that you almost think that she is too young for it to be true.

You can use "awfully" for bad things or good things:

It's awfully expensive, don't you think?

You're an awfully nice guy, but I just don't think I'm ready to start dating again.