Just trying to establish the time when the term came into popular use in the US or elsewhere. Google didn't tell me.
To work out (a sum) by means of figures; more widely to estimate or calculate; hence to work out; make out. Chiefly U.S.
1833 As I said before I'm stump'd about that Bank of US; and I want you to help me figure it out. OED
I found these early examples with Google Books:
"Well, Mr. Clerk, ... you have been figuring it out for me on paper; has he got it right?" "Yes, Madam," said the clerk, taking his long calculation out of his pocket, "he has got it exactly. And, I venture to say that there is not another man on the island that can do it in two hours." ref. (1838)
"Sure enough, brother Cap, your question is more easily put than answered. Is there no such thing as figuring it out by navigation? I thought you salt-water mariners were able to do as small a thing as that. I have often read of their discovering islands, surely." James Fenimore Cooper, The Pathfinder (1840) p.189
As Andrew Leach points out in his answer, there is a broadening in meaning of to figure out from to work out/understand via calculation to its more general meaning. This is similar to what happened with the verb to reckon centuries earlier.
The meaning becomes more figurative—wait, no, less "figurative" :-)
The thing to do is to use Google ngrams which would indicate 1885 for the first appearance in print.
One can then tweak a Books search (using the buttons at the bottom of the chart and changing the dates in the URL) to find an occurrence in 1867:
I can't be positive about it, but I believe I did figure it out.
That usage refers to calculation, because the next question is "Was your estimate higher or lower than £55000?" The verb appears to be similar to "put it out" — the result drops out of a calculation involving the manipulation of figures.
It's a short hop from calculating a value to working out how a particular situation arose. By 1918 the phrase appears in a car workshop manual:
Find if it is ignition, carburetion, cooling, or just what the trouble is, and then figure it out the best you can before proceeding.