But how exactly do the voltage and current propagate through transmission lines? 

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Here I explain by animation how exactly the voltage and current propagate through the transmission line, How reflections affect the signal transmission, The concept of standing wave ratio and motivate the viewer to smith chart.
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These animations are made using manim. A community-maintained Python library originally created by grant sanderson (3blue1brown) for creating mathematical animations.
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commons.wikimedia.org/wiki/Fi...
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0:00 Introduction
1:40 voltage and current waves
2:09 what is complex exponential function (the forward and backward waves)
2:48 the standing wave pattern (the first perspective)
4:18 the standing wave pattern (the second perspective)
6:29 the standing wave pattern (the third perspective)
7:39 the standing wave pattern (the fourth perspective)
9:17 the matched load: standing wave ratio (swr) of one
10:08 unmatched load: standing wave ratio (swr) between one and infinity
10 :45 full reflection of the wave infinite standing wave ratio (swr)
11:12 impedance transformation and smith chart
14:23 transmission line delays the signal and my change the amplitude periodically while propagating if the load isn't matched