Whenever your Browser finds a solution to this puzzle, it sends it over to our servers where it is verified. Once your Browser has submitted enough valid solutions, our servers will allow you to submit the form.
This essentially works like Google's reCaptcha (you know, the one where you have to “proof you're human” by selecting all street signs from a bunch of images). This captcha however lets your Browser do all the work.
Depending on your Computer, solving these mathematical puzzles can take quite some time. The faster your Computer is, the faster the Proof of Work Captcha will be solved.
Smart phones or tablet's are especially slow at this. If you want it to run faster, use a Laptop or Desktop Computer.
The mathematical puzzle is the hashing algorithm for the Monero Blockchain. Monero is a Crypto Currency, not unlike Bitcoin. We use Monero, because its hashing algoritm (Cryptonight) runs especially well on consumer CPUs.
Essentially, you're mining XMR - the currency for the Monero Blockchain. With each hash you solve, you create a tiny bit of XMR, less than one one-thousands of a dollar cent. But it all adds up.
The proceeds go to the authors of the Website and help to cover their costs.
You can embed this captcha in your own website, too and let your users mine XMR for you. Have a look at coinhive.com for the details.