How does the universe expand?

My belief is you are asking how the universe is expanding, as in if the universe is everything what is it expanding into.
The best way to answer this is to imagine a 2 dimensional universe. If you wrapped that universe around itself and join it the surface could look like a sphere. Now if you inflate that sphere like a balloon, it expands. Anything 2 dimensional being living on that surface would not see what the sphere was expanding into just that it was expanding. The reason the 2 dimensional being couldn't see it is because the sphere is expanding into the 3rd dimension.

Now all you have to do is add a dimension, and then when you ask how a 3 dimensional universe is expanding the answer is in the 4th dimension.

The evidence for this model is gravity. If you press with your finger into the surface of the sphere you create the equivalent of gravity. In that case gravity is actually bending the space. In our universe we see gravity being able to bend space, when we look at light that travels near a very large body (like the sun) we see the gravity of the sun can actually bend the light. Furthermore, the way that black holes behave, gravity so strong that light cannot escape, it further demonstrates the model.