It means "to endure something" or "to tolerate something". The idea is of accepting something that you don't want to accept, e.g.
1. I'm not going to put up with her bad cooking any more = I'm not going to accept her bad cooking patiently any more.
2. I have to put up with a lot of abuse from members of the public = I have to endure a lot of abuse, etc.
3. I don't know why you put up with it = I don't know why you accept the situation (instead of trying to change it).