When you were in your 1st year of life, you could say that was year "0".
Things get confusing (for many people) when numbers start to get bigger: The 20th century were the years 19xx (mostly). Did the 2nd millenium end on New Year's Eve 1999 or 2000?
For mathematicans or sientists in general (as mathematics is nowadays a big part of every sience) it is quite natural to count from zero.
Programming languages were developed by sientists mostly. So there is the historic background.
About the only language coming to my mind with 1-based arrays is BASIC. But looking at the BASIC mainline nowadays (VB.NET that is), Microsoft has also evened that out. While you have the possibility to declare arrays with arbitrary lower and upper bounds, a standard array will index its elements from zero.
Also going for the "natural" argument again. As an example in Perl look at the possibility to access the last array element via $array[-1]. This seems just right, concidering that the first element is $array. Conversely with $array as the first element. By allowing round-robin-indexing you end up with $array as the last element. Which looks weird, in my eyes at least.
Update: Some minor typoes fixed.