This doesn't always refer to a physical exit. In fact, they way it is used being "gave her a way out" makes me think it is not talking about a physical exit.
In any tough situation, one from which a person wants release, a way out can be offered. This way out can be any alternative that allows someone to escape from obligation, or allows them to get away from the tough situation.
It would be helpful to know the context, but whatever it is, look at the context where you got that. Most likely "she" owed a debt, had to face consequences for something, or had some very unpleasant task to perform. "He" probably offered an alternative so she would not have to pay her debt/face the consequences/perform the unpleasant task, etc. The alternative he offered her was the way out. By offering this alternative, he gave her a way out.