Physics behind signal reflections and series termination
With electrical transmission lines, it all has to do with the speed of light being finite, thus so is the speed of EM propagation in a wire. You can think of a wire as a long series of infinitesimal capacitors (connected by infinitesimal inductors). If you start charging the capacitors at one end, you have to keep pumping charge into the wire to charge more and more of the infinitesimal little distributed capacitors down the wire (at the speed of EM propagation).
Then a problem occurs when you get to the end of the wire.
If shorted, the last capacitor can't charge, and it in turn then discharges the capacitor one back. This discharging of capacitors then travels back to the source (at the speed of EM propagation) as a negative voltage reverse wave that cancels out the forward positive wave.
If open, remember there is still charge being pumped into the line. It has to go somewhere, so the "momentum" of the current (due to distributed inductance of the wire) charges up the last capacitor to double the voltage of the rest of the line capacitance. But this last capacitor is connected to all the ones before it. So this wave of charging to double the voltage then propagates backwards on the wire as a positive reflection wave.
Somewhere in between a short and an open at the end is a nice medium value of impedance which can absorb the wave of current hitting the end without either under or overcharging that last bit of capacitance at the end. Thus leaving no change in voltage to propagate back along the transmission line. That terminating impedance just happens to be the characteristic impedance of the transmission line, which is determined by the distributed capacitance and inductance of the wire (and its surroundings: permeability, ground plane or return geometry, coax shield, et.al.)
Pretty much the same thing occurs with air pressure waves in a tube or pipe (organ), or when whipping one end of a rope sideways. Different rope waveforms occur depending on whether the rope is tied or loose at the opposite end. etc.
(Added: For transmission lines where there is a discontinuity or mismatch in impedance somewhere in the middle, you can think of it as a superposition of two transmission lines, one line with no discontinuity over the total length, plus one shorter line with an open or short at the discontinuity. The reflection will relate to the ratio of the two superpositions.)
The impedance of a transmission line, in ohms, is the ratio of voltage wave and current wave that travels down the line. For a 100 ohm line for instance, a 1 volt wave will always be accompanied by a 10mA wave. Intuitively, the current wave delivers charge to the parts of the line that have to 'charge up' to the voltage of the voltage wave.
If the 100 ohm line is now connected to a 50 ohm line, the ratio of voltage and current waveforms in that line will be different. The junction between the lines cannot physically support two voltage and current waves with different ratios. A third wave is therefore generated that 'takes up the slack' between the mismatched waves, and is reflected back along the source line. Note that the reverse travelling current wave will subtract at the junction, whereas the voltage waves will add. It's therefore always possible to find an amplitude and phase of a reverse wave that will match any two lines.
If the line is loaded, or sourced, with a resistor equal to the impedance, then a voltage/current wave arriving at that junction finds that it can continue to propagate with just the right ratio of voltage to current, and so no reflected wave is generated.
out on a limb here ---- the (energy, as Tim Wescott writes) differential equations are used, with boundary conditions that require the sudden appearance of waves traveling in the reverse direction IF that boundary is not exactly Zo; the energy is preserved in the new mix of voltage/current values for each of the forward and (now) reverse waves.
regarding "series terminations" --- a series termination, best used with the resistor installed at the SOURCE end of the transmission line, exploits reflections in its operation. Initially the line voltage is only half the Source voltage because of the voltage division of the lumped resistor driving the Z of the line; any circuit monitoring the line will see only Vin/2 and that often is a FORBIDDEN VALUE for logic circuitry; however, at the far end, the receiving end, the math tells us that unterminated receiving end HAS A REFLECTION, and the math tells us the voltage doubles as part of preserving the energy. Thus ONLY at the far end, the receiving end, will a useful full amplitude waveform be created.
At all other points along the line, the voltage will be HALF for some time, and then the reflected energy doubles the voltage. This doubling occurs, eventually, at all points. In general, trying to extract data from this 50% then 100% waveform is a bad idea.
Only at the far/receiving end does a safe-to-use waveform exist.
On the other hand, the use of series-at-the-source termination will reduce overall power consumption.