Think of "figure out" as finding a result for a problem, or in other words, "solving" either a problem, a conflict, or an uncertainty.
The easiest example to understand is with a math problem. You are given numbers (or in this case, figures), then you use those figures to solve, or "figure out" your problem. This can be used outside of math as well, here are some examples:
I'm not sure what I'm going to do to earn the money for rent this month, but I will figure something out.
Something doesn't seem right, so I'm going to figure out what's going on.
Person 1: "I don't know how I feel about our relationship anymore."
Person 2: "Well, figure it out!"
So in all of these it's used either as a way of solving a problem, or (in the second example) finding information to solve a mystery. "Figure out" is commonly used in spoken English, so it is an important idiom to understand. :)