If you know what an FFT is you know that it's very often used for signal processing. You probably also know that convolution in the time domain is the same as multiplication in the frequency domain, and vice-versa.

So now imagine a multiplication of two numbers, say 12345 * 67890. Imagine each number is a signal where the digits are samples. Multiplying like you were taught in third grade is effectively convolving one signal by the other.[0] Do this with pen and paper to see it clearly.

So if we call normal third-grade multiplication "time domain convolution," we can imagine converting both signals (both numbers) into the "frequency" domain with two applications of the FFT, then multiplying their frequency domain samples component-wise (which is much faster than convolution) and then taking an inverse FFT of the result. That's the algorithm in a nutshell.

It would buy you nothing on the numbers above because they're much too small. But for numbers with thousands of digits, it's a big speedup from O[n^2] to roughly O[nlogn].

[0] I'm oversimplifying. It's actually polynomial convolution and you have to do carries. That's slightly harder than signal convolution, but the idea is the same.

I gave a quasi-explanation in response to lamename's comment. If you want some search queries, learn what convolution is, then observe the connection between discrete convolution and polynomial multiplication, then learn about the convolution theorem.

If you have CLRS 3rd edition, check out chapter 30