The 'bloom to her cheek' means that she blushed slightly, like a (red) flower blooming on her face.
I'm not sure about the other reference - you don't say where it comes from. However, I would guess that the meaning is to do with wooing a person (usually a man woos a woman, pays her a lot of attention, makes it obvious he would like a relationship with her) and winning means to succeed in the wooing. So, maybe he wooed her, and then proposed marriage and she accepted? I hope this explanation makes sense in the context you found it.