Because at the dawn of civilisation, ancient farmers realised that the Sun rose at different places on the horizon every day and that the position of the rising of the Sun seemed to have some connection with the rains and ability of land to produce food. They also noticed that the moon moved and waxed and waned...
These numbers arise out of approximations of nature: 360 is pretty darn close to the 365 days it takes for the Earth to revolve around the Sun, but has the advantage of being much more readily divisible into smaller parts. The prime factors of 365 are five and 73: that's not very convenient for dividing up a year into smaller chunks; 360's prime factors are two, three, and five. That creates a lot more flexibility. (Throw in a couple of sacred holy days throughout the year and you're good to go!)
Given the number of lunar cycles in a year, it makes some sense as to why night and day might be divided into two sets of 12 segments.
About the 60s: it's because the people who created divisions of time, and before that, divisions of angle, used base 60. Ptolemy (c90-c168), for instance, wrote his table of trigonometric functions in base 60, not just for the angles, but for distance measurements, too.
Ptolemy, and the Greek astronomers before him, used base 60 because they learnt it from the Chaldean astronomers, late Babylonians, who naturally used base 60 because that's how they wrote their numbers. The Babylonians, the Assyrians, the Elamites, and the Sumerians all used base 60 since the beginning of writing over 5,000 years ago.
David Joyce, Professor of Mathematics at Clark University, Massachusetts
Numbers like 12 and 60 are actually far better bases than 10; 12 can be divided evenly into one, two, three, four, or six pieces. Sixty can be divided evenly into one, two, three, four, five, or six pieces. Ten can only be divided evenly into one, two, or five pieces. Since dividing something into three or four pieces is a lot more common than dividing them into five pieces, numbers like 12 and 60 are superior. (Imagine if, instead of "15 minutes," a quarter of an hour was "two and a half decents".)
The only reason we use base 10 for everything else is because, at some time far in the past, people decided that counting on your fingers was a great idea. This was a bad decision, but it's stuck with us for a long time.
Daniel Mclaury, PhD student in mathematics
These answers all come from quora.com, the popular online Q&A service. Ask any question and get real answers from people in the know