First, a tire's rolling resistance is not caused by friction to the road surface. It is caused by the internal friction of the material.
There are primarily three things that affect RR. The amount of deflection (movement of a portion of the tread as it moves in and out of the footprint - which is basically a load vs inflation pressure thing), the amount of material being deflected (basically the amount of tread rubber), and the nature of the material (basically the properties of the tread rubber).
Here's a more through explanation:
Barry's Tire Tech
Now the speed effect is because there is a thing called a standing wave that generates more movement - and hence more energy consumption. The faster you go the greater the energy consumption.