Why does February gain a day every four years?

The Earth takes about 365.2422 days to go round the Sun, so a 365-day calendar loses a quarter of a day a year. Adding a day every fourth year very nearly fixes it, and Julius Caesar's calendar did exactly that from 45 BC.

Very nearly is not the same as exactly. The true figure is slightly under 365.25, so the Julian calendar gained about eleven minutes a year - trivial per year, and ten full days by the sixteenth century, by which time Easter had drifted noticeably away from the spring equinox. The Gregorian reform of 1582 dropped the extra day in century years not divisible by 400, which is why 1900 was not a leap year and 2000 was.

Adopting it took a long time and caused genuine anger. Catholic countries switched in 1582; Britain and its colonies not until 1752, by which point eleven days had to be removed, and the calendar went from 2 September straight to the 14th. The stories of riots demanding those eleven days back are mostly exaggerated, but people with rents, wages and contracts falling due had entirely real grievances. Russia held out until 1918, which is why the October Revolution took place in November.