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.

Perhaps it would be useful to think of it differently. Instead of shoving electricity around, pretend that you have a very long pipe, some water (at pressure) and a valve.

If you turn on the valve at one end of the pipe, it will take a certain amount of time to pressurize the pipe and force the water through. At the other end, the water will eventually come out, but as a slow increase in flow, followed by a slow decrease.

If you turn the water on and off quickly enough, it will simply appear at the other and as a moderate flow.

As mentioned elsewhere, in electrical terms, this is due to the capacitance, and inductance of the long cable. The longer the cable/pipe, the greater the apparent smoothing effect. The reasons (physics) and math may be different, but the results rhyme, and hopefully that will help you understand.

Infinitely sharp edges in the signal need infinite bandwidth to transmit which is not possible with real world wires. Taken a long enough wire, it passes only low frequencies and high frequencies are attenuated, so the sharp fast edges get smeared into slower wavy edges and thus you need to send longer pulses to see the voltage to slowly rise up to proper detection level at the receiving end. The signals degrade as each unit length of the copper wire can be thought as a series resistance with inductance and it also has stray parallel capacitance and leakage resistance to its surroundings, otherwise this model is known as a transmission line. The series resistance with parallel capacitance is a low-pass filter. The capacitance and inductance forms what is known as the characteristic impedance of the transmission line, and if the transmitting and receiving ends do not match the transmission line impedance, a voltage pulse will reflect back to wire to some extent so it ping-pongs back and forth in the wire until it settles. As the voltage pulse travels in the wire at roughly two thirds of the speed of light, all this how much the signal degrades when it travels through the transmission line or keeps bouncing between the ends of transmission line determines how fast signals you can transmit and how far you can transmit them without being too degraded.