I know of two ways to explain this phenomenon. One uses lumped models, and the other uses quantum physics. I prefer the lumped model. I have drawn the lumped model case where the end of the transmission line is terminated in an open circuit. Note that the lumped model is a way of describing the transmission line with components such as R, L, and C.
1) The wave from Vs begins to propagate down the transmission line
2) The first capacitor is charged
3) The second capacitor is charged
4) The third capacitor is charged, but since current cannot flow through another branch (because of the open circuit), the inductor and its unstopping current charge the third capacitor beyond Vs.
5) Since the potential on the third capacitor is larger, now current flow will eventually reverse direction and flow back towards the source, charging the subsequent capacitors.
Now note that if the transmission line were terminated in a short, the current would flow through the short and back towards the source on the bottom of the circuit. This would cause each capacitor from right to left to discharge in turn, which causes 0 volts to be seen on the line. So, with both short and open explained, anything else is a case in between these two extremes.