There are a couple parts to this question.
First of all, February has one more day in a leap year because in a leap year one day must be added to the calendar. This is because the Earth actually orbits the sun in roughly 365.25 days. So every 4 years we need an extra day. Otherwise, the dates would stop corresponding to the right seasons and we would have, for example, Christmas in the middle of Spring in the Northern Hemisphere. So, to keep the calendar date matching with the seasons, we have to have leap years.
The second part of this question is why February has so few days. My understanding is that it used to have 30 days. But then Julius Caesar wanted another day for "his" month of July. So he took one from February. Augustus Caesar didn't want Julius's month to be longer than "his" month of August so he took another day from February.
Because the length of days and the calendar year are based around the rotation of the Earth and how long it takes the Earth to revolve around the Sun one time. This takes approximately 365 days to do, or 365 rotations of the Earth. But since those two measurements are not directly related, they are a bit unequal.
So it actually takes about 365 and one quarter days for the Earth to revolve around the sun. After four years, add those one quarter days into one full day, and add it on to the shortest month, February, to give us a "Leap" year every four years. We're basically "leaping" to catch our calendar up with the actual revolution of the Earth.
Leap year is basically a year with an extra day in it, February 29, and occurs once every four years. The reason for this is to keep the seasons, astronomical events and time differences in sync. This has to do with the Gregorian calendar, which is used almost everywhere in the world. Leap Year is directly related to the Earth and how many times it orbits the sun in one year. There are not actually 365 days in a year, but 365 1/4 days in a year. So basically, four quarters would make up an extra day, hence adding a day every four years.
because in one year, we have 365.25 days.
while we have Leap year every 4 years.
means, the additional day in Febuary, has something to do with the earth's rotation.
.25+.25+.25+.25 = an Additional 1 day.
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so during leap year, we have 366 days:)
February used to be the last month of the Roman calendar. For this reason it was the one that suffered tinkering when Julius Caesar's astronomer Sosigenes created the Julian calendar to account for leap years. Whwn Augustus Caesar renamed the month following Julius's (July) after himself he pinched a day from the end of the year to give his month the same number of days as July (i.e. 31), thus shortening February.
There are 365 1/4 days every year. We only celebrate 365 of those, leaving the poor quarter day abandoned. BUT avery 4 years, each of those quarter days add up to be one whole day--making every fourth year have 366 days. It was decided to add that day at the end of February since that month was shorted a few days anyway!!