The process is complicated but this should give an intuition into what's going on. Imagine you have a regular camera, but with motors to move the sensor half a pixel in any direction.

Instead of taking one image this camera takes four: one centred, one shifted half a pixel right, one shifted half a pixel down, and one shifted half a pixel right and half a pixel down.

We can then take the centred image, make it double the size, spacing out the pixels like so:

xxxx               x x x x 
xxxx      ____\    
xxxx          /    x x x x 
xxxx               
                   x x x x

                   x x x x

Then we can fill in the gaps, using the other shifted images, 1, 2, and 3:

x1x1x1x1
23232323
x1x1x1x1
23232323
x1x1x1x1
23232323
x1x1x1x1
23232323

Leaving us with a image of twice the resolution. Interestingly enough there are cameras than employ this technique - such as the Hasselblad H4D-200MS (sorry if you have to ask how much you can't afford one).

Superresolution with a standard camera is a bit more complex as when you have uncontrolled camera or subject motion you don't get anywhere near an exact half pixel shift, but unless you are extremely unlucky your shifted image will be some amount offset from the original. By combining enough images you will get a very irregularly sampled image (with pixel samples that don't fall onto a grid) but one which can be interpolated (by tracing lines between samples to guess a result that does fall on an exact gridline) into a regular image.