Answer:
To figure means to calculate (as in, do figures), and in this case is extended to mean "think through", a meaning for "to figure" also found in "figure it out"( think about it until you understand it).
When someone tells you to go figure, it literally means they want you to think about something. When the "something" isn't specified (the phrase is used at the end of another phrase or by itself), it refers to whatever has just been said, and indicates that the speaker thinks there's something funny about it (though often it's black humour and it's depressing as soon as you stop laughing), and almost always about someone (or mankind as a whole) being stupid or Murphy's law applying. Can often be replaced with "which was to be expected" or "can you believe anyone in his right mind would do that?"