Experimental Proof of Time Dilation?

As long as you're good on the time dilation part, the explanation is fairly easy. From the muon's point of view, the earth is traveling towards it at relativistic speeds. The distance from the upper atmosphere to the earth's surface shrinks in the direction of travel so the muon "thinks" it's actually less than 25 miles. Thus, while the muon lives a "normal" amount of time in its own reference frame, it doesn't have to travel as far as it would if it were traveling slower and therefore can reach the surface. The length contraction factor, gamma, is calculated in the same way that we calculate the time dilation factor.