Also, as someone else mentioned, it might keep beginners out, which could be what they want? Seems like a bad idea, but it's not my board so...
It's a topical captcha that the target audience of the site will recognize and most likely enjoy.
It's significantly different from any other captcha so that automated untargeted bots will not be able to beat it
It has a high enough entropy that untargeted bots that repeatedly attempt to guess the answer will fail
People without a chess background most likely have to think about the answer in a way that makes buying human workers a less attractive option (maybe)
Anyway, the same effect could be obtained by writing a "captcha" that simply asks: "Click on the queen", a fixed chess board picture. They are assuming no one targets them, so it's not so important how hard it is. (until someone targets them)
Most of these captchas are very easy. If the targeted king was easier to spot (e.g. marked red) I'm sure I would solve them faster than the omnipresent reCaptchas (even knowing that I only need to type the less legible word).
And as somebody else pointed out, maybe they don't want new topics asking what are the possible moves of a queen.