If one sends short bursts of electric current that abruptly turn on and off, why does the receiver receive a smoothed-out current?
Your cable behaves like a low pass filter, which means that high frequencies are getting damped. The longer the cable, the stronger is this effect.

Impulses have, due to their fast rise and fall, pretty fast frequency components. 
If these high frequencies are damped, your impulse "smears" over time and you get the desired result you posted in your question.