Its said to people who are begging for sympathy and the person they are seeking sympathy from has none for them. I think you're referring to the Justin Timberlake song? He never said outright that the song was about Britney Spears but in the song he's talking about a girl who cheated on him now she wants to play the victim and he saying he doesn't feel sorry for her so she may as well keep crying or cry him a river.

When someone says "oh cry me a river" it means they are not sympathetic to that persons sadness. Go a head and cry me a river, get over it. You are making a bigger deal than it is. Probably the person crying is crazy for the one saying cry me a river, and they could care less about that person and what they are upset about. This is my best guess.