What exactly does the expression, "that's rich" mean?

Sarcasm for extreme.

It’s a good thing. A compliment. Being rich is always a good thing, something all strive to be, so it’s saying that whatever is the subject of the remark has achieved to be the best of the best in the class.

Rich, in this sense means highly entertaining or amusing; preposterous or outrageous; although less common today it was often found in the phrase ‘a bit rich’ or the ironic comment ‘that’s rich!’ to express ironic amusement or indignation at something. The root sense is one of rich as meaning plentiful or abundant.

http://www.quora.com/Where-does-the-phrase-Thats-rich-originate

@bkcunningham Interesting link, it seems like one of those words that can mean both one thing and sometimes it’s it’s opposite. I have always understood it to be something sort of sarcastic, the second two, really the third, of the meanings. “That’s rich!” has always meant that’s “That’s preposterous” or “That’s stupid.” I guess it’s a matter of the tone of the speaker maybe. I suppose just about anything can be said sarcastically, but that’s the way I’ve always heard it used, and I think maybe I’ve only heard in in movies, I’m not sure I’ve ever heard it in a real conversation.

Another phrase with multiple meanings to ponder. It make English complicated, even for native speakers. A fourth meaning, said with a different inflection: “That’s got a lot of butter, sugar and chocolate etc. in it!” I don’t think I’ve ever used it, except that way. Have you? Maybe that’s the origin of the phrase?

The sardonic “that’s rich” is really hard to explain. It’s a bit like saying that a bit of bull. Or it’s ironic. Or that you don’t believe the person. It’s like saying it’s too much to believe. It is a statement of a kind of disbelief—not that the person is completely lying, but they are definitely stretching the truth in some way.

Not necessarily. One might use it to explain an excess of something may not be required.