This is by far one of the most challenging and popular HSM questions on the Net. Proofs are, countless discussions about it in math forums. The answers only led to two theories, which Wikipedia does a great job at summing up.
The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers.
Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system.
The two theories seem very legitimate. With some background search I came up with the fact that the use of astrolabes by Persian astronomers (which used circles as its fundamental "measurement medium") could actually lead to invention of such system by them.
Nevertheless, Babylonians are known to be the inventors of sexagesimal system and additionally, their more "recent" studies were using such system in the advent of trigonometry.
A deeper study also indicates that Persian people could've actually be influenced by Babylonian "culture" to make use of their sexagesimal system. This makes logical decision harder: Maybe it was Babylonians who invented 360 degrees in a circle, or it could be Persians that did so with the sexagesimal system that was introduced to them.
Unfortunately, I'm not able to cite my non-English sources, but it would be great to hear an expert's advice on this.
Who were the first to use degrees in a circle and divide it to 360 parts?