The short answer to why we use it is that we inherited it - base 12 of hours/months from the Egyptians and base 60 of second and minutes from Mesopotamians (who got it from the Sumerians).
Egyptians used base 12 a lot for a similar reason that we use base 10 a lot. We use 10 because we have ten fingers, and they used 12 because one hand has 12 knuckles (they'd count on one hand). But it was handy because there are 12 lunar cycles, so it helped keep things more consistent.
Base 60 is also handy because 60 is first number divisible by the first six counting numbers and by 10, 12, 15, 20 and 30. If you use 60, you have options! Note that we also use 60 for angles and dividing up the globe.