Because physics. The voltage on a [linear] transmission line can always be decomposed into the superposition of two waves travelling in opposite directions, at equal velocity.
When you measure the voltage and current at a low frequency, it may seem stable, and arbitrary (i.e., V and I unrelated to Zo), but when we apply this identity, we find it's the equilibrium between reflected waves.
This is perhaps easier to see when we charge up a transmission line with a step current. Say we have a 10V, 1A source at one end, and a 10 ohm load at the other end. The line is Zo = 100 ohms and t_o = 1us long.
When we apply the step voltage, at first, 10V / 100R = 0.1A is drawn, and this continues until one electrical line length.
At 1us, the wave reflects off the resistor; we can draw a voltage divider between Zo and RL, and find that 90% of the wave gets reflected, and 0.1A is delivered to the load. (The source is still delivering its steady 0.1A.)
At 2us, the reflected wave returns to the source, and its amplitude superposes with the first wave. Because the wave was reflected out of phase, more current is drawn; now the source delivers 10V and 0.19A (give or take). No voltage is dropped from the source, so the wave again reflects.
At 3us, the load rises to 0.27A. At 4us, the source rises to 0.35A, etc. The rise continues, but it does seem to be slowing with each cycle. The wave that's always getting reflected back and forth, is decaying as it goes -- it's doing the work of ferrying source and load signals back and forth, communicating that the load is still a lower impedance than the line, and still able to draw more current. It takes a relatively long time to do this -- the load is 1/10th of Zo so the time constant is around 10 line lengths (round trip), and it will be 99% of the way there after say 60us.
Which to human eyes, still looks instantaneous, so we can easily overlook it if we don't have an oscilloscope watching carefully. Also, a practical note: 1us long cables are, well, very long. 200 meters or so. That's a big spool to just sit in the lab and measure. Real transmission line is lossy, meaning the wavefront does not stay sharp, but its leading edge drops and smears out, rounding off the wavefront and increasing its rise time. The effect of this is, you may see the stepwise current increase for a few cycles, but pretty quickly the steps will blur together, and it will look like an ordinary RLC circuit charging up.
Lastly, I probably should've just gone with the other case, RL > Zo, because 200 meters of line will probably have more than 10 ohms of DC resistance, to begin with. Which means you can't run this test without also taking account of the line's losses, which is annoying. (A good lesson for practical experiments, surely, but not an impressive one when you're expressing things from an ideal basis.)
Anyway, once everything is settled, and regardless of the line's high frequency loss or dispersion, the result is, eventually, the wavefronts decay to zero, and you are left with the superposition of source and load waves, constantly being launched from each end, crossing by each other, which happen to match and oppose perfectly to give the observed voltages and currents at both ends. You don't see it as waves in motion, because the waves aren't carrying motion anymore; but they are, in a sense, the more accurate way to see things!
For practical purposes, by the way, we're much more inclined to use a low-frequency approximation to full transmission line theory. We can convert a line to a series inductance around t_o * Zo when we use it at R_L < Zo, or a parallel capacitance around t_o / Zo otherwise. (We may need to model it as both L and C, or as an arbitrarily long LC ladder network, if we're interested in approximating its impedance and time delay properties for Z ~ Zo.) Whenever you see mention of stray inductance or capacitance, this is actually what's happening: we're approximating a real electromagnetic wave structure with low-frequency equivalents.
Anyway, even for good old fashioned telephone purposes, as well as DSL, DOCSIS and others: the fact is, we can decompose the line voltage into two propagating components. When we do that, we can devise networks that assist us in separating them, and understand the conditions necessary for their use. In telephony, the key component is the hybrid transformer. At radio frequencies, we might do it per band, with resonant circuits (or at very high frequencies, transmission line segments again!), or use a wideband hybrid or bridge to do it for all frequencies of interest.
Or heck, with so much computing power available on ICs these days, a lot of this may be absorbed directly into the DSP domain. All we really need to know are the voltage and current we're delivering to / drawing from the line, at any given instant; and the impedance of the line. If we know what we are sending (say from a power DAC), then we know what the line should be due to our transmission, and whatever difference is left must be due to received signals. Read with ADC, subtract and process.
Also, when bands are split by frequency, we can simply use a diplexer to separate the frequencies. If DOCSIS upstream is < 50MHz and downstream is >, we simply use a diplexing filter to separate these into high and low bands. The filter sharpness is the isolation between transmitter and receiver, and we can further design the receiver to reject transmitter frequencies so that it receives only the relevant signals. Done on a broader scale, this is FDM (frequency division multiplex).
Tim
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