Used in a strict sense it means performing very well. I think it may come from horse racing, and the form (fitness rating) of the horses. A horse might be describes as flying if it is beating the others by a long way. It can be used to describe anyone at the peak of their performance.
However, in the case you mention it means that her personality was very lively from the effect of a lot of alcohol. She was probably very talkative and opinionated. It may be being used sarcastically here, if the speaker was not impressed with her drunkenness.