We have the Babalonians (and possibly Egyptians partially) to thank for this.
Instead of using base 10 as a numbering system like we do, Babalonians used base 60. This made some sense to them, perhaps simply because it is easy to divide it by many numbers to produce another integer. I.e., 60 can be divided into by 2, 3, 5, 10, 15, 20, 30... so it makes it easy to split up, if you're working out nice ways to split farm land or other things.
They also had an interest in the heavens, so naturally used these same units. Splitting 60 minutes into 60 seconds, that type of thing.
It's also the same reason why there are 360 degrees in a circle. They liked the number 6 too (given their numbering system), so perhaps used 6 lengths of arc in their circles, each of 60 degrees, to make 360 degrees in total.
Google for bablonian 60 degrees to find out much more.