Why does it stay twilight all night during the summer at far northern latitudes?

At far northern (and far southern) latitudes during the summer, the sky remains light much or all of the "night". In northern Russia, these times are sometimes called "white nights", because the sky never really gets dark. The Sun sets, if it does, far north of west, and doesn't go much below the horizon, and after an hour or two, it rises again far to the north of east.

A personal tale; I was in the US Navy, and our squadron was deployed to Keflavik, Iceland, only a couple of degrees of latitude south of the Arctic Circle. It was late in the day, so we went to the officer's club and had dinner. Then a few drinks, then watched a band play for an hour or three, and a few more drinks..... We left the club at 2AM.

I want you to know, there's not much worse than having the beginnings of a hangover and to walk out of the club at 2AM and get smacked in the eyes by the already-rising Sun.