How did they explain that?
Generally, there was no need to because people who knew about that typically accepted it as part of the evidence for a spherical Earth. (Not always actually knowing there were geographical zones where it is daytime for half of the year then night for the other half (approximately, since strictly this only happens at the poles), but deducing it from the variation in the elevation of the Sun with latitude.) Other evidence included the way that things appeared over the horizon, e.g., ships being visible mast-first, then hull, the horizon being further away when viewing from higher elevations, the roundness of the Sun and Moon, the Earth's shadow during lunar eclipses, etc.
Old-time people who believed the earth was flat either didn't know about this (and therefore didn't need to explain it) or had heard about it but didn't bother to think about what it meant (and therefore didn't bother to explain it).
The most common modern Flat Earth movement explanation is the one from the great founding text of the movement, Zetetic Astronomy. The Earth is assumed to be a flat plane, with the north pole at the centre, and a rim of impassable Antarctic mountains along the outside. The Sun is assumed to be relatively close, and at the equinoxes travels in a circular path above the equator, with its light just reaching the north pole. In summer, it travels in a smaller circle, so its light will always reach the north pole, and in winter, a larger circle, with its light never reaching the north pole:
No adequate explanation is given for why the Sun's light only illuminates a finite patch of the flat Earth. Long nights in the south are explained by the same explanation. Long days at the south are not, and the author simply "explains" long southern days by denial. A variety of "explanations", still including denial, are offered by present-day proponents of a flat Earth.
Reference:
"Parallax" (Samuel Birley Rowbotham), Zetetic Astronomy. Earth Not a Globe!, Simpkin, Mashall, and Co., London, 1865.
Two online versions; the chapter linked above is from the latter:
People knew the earth was a sphere going back to classical antiquity. I wrote about this here before: https://www.reddit.com/r/AskHistorians/comments/3uufv2/when_did_people_understood_the_concept_of_time/