From numberphile:

Key insight:

- A complex number, when multiplied by itself (x^2, x^3, etc.) will scale the magnitude and add the angles
- If we have a circle of possibilities (same magnitude), we could have many angles along that circle. For example, if 60 degrees is our angle, and we got there in 3 steps, then 20 + 20 + 20 would work (1/3 * 60), along with (60 + 360) / 3 = 420 / 3 = 140, since 140 + 140 + 140 = 420 = 60 degree angle. Along with (60 + 360 + 360) = 780 / 3 = 260. Three different angles that, when combined 3 times, give us 60 degrees.
- So, we show that a root exists, and there are N points that (when multiplied) would get us to that angle.