Can someone please explain this phenomenon? A solid explanation seems to be missing from the entire internet.
You know, the first time someone told me this, I was absolutely certain they were confused, ignorant, or otherwise mistaken, and I told him he had his facts wrong. To be precise, we were talking about an observer on the equator. I maintained that the Sun would rise due east and set due west, all year round. He said no.
What gave me pause, though, is that this person was far from ignorant, so maybe the other two appellations didn't fit either. I went home and tried to construct a convincing proof of my side, and in the middle of the night a very simple and convincing proof did come to mind. Except that it was a proof that I was wrong.
Imagine that you're on equator at sunrise on the summer solstice. Due east is along the equator. But the plane of the equator is tilted relative to the Earth's orbit, and the Sun is well above (north) of it. You cannot see it due east at sunrise. It has to be $23.5^\circ$ north of east. (Remember, this is for an observer on the equator.)
By the same token, the Sun must rise $23.5^\circ$ south of east on the winter solstice, and gradually transition between these extremes during the year.
[For this explanation, it's helpful to ignore the slight apparent $1^\circ$ motion of the Sun throughout the day].
The way I like to think about this is imagine standing at the North Pole in the summer time. Over the course of a day, all you're going to do is spin in a circle, and the Sun will make a circle around your zenith. When you're at the one of the poles, your definition of "up" and the Earth's rotational axis are the same. So, of course, the Sun will appear in every direction throughout the day, including directly North.
If you then move 20° to the South, you have rotated yourself 20° from the Earth's rotational axis. So the Sun will still make a circle in the sky, but it will no longer be centered on your zenith. Instead the center will be shifted by the amount you rotated yourself (with respect to the Earth's axis).
If you then move a little further to the South, that circle will begin to intersect the horizon. But only a small piece of the circle is removed.
To exemplify this, the images below show the path of the Sun through the sky at 90°N, 70°N, and 50°N near the summer solstice. You can try different values, and other celestial objects using this Sky Path page.
Path of Sun viewed from North Pole:
Path of Sun viewed from 70deg N.:
Path of Sun viewed from 50deg N.: