Tell me again what century we are in...

There's no year "zero". January 1, 0001 to December 31, 0100 is the first century. The first century ENDS on December 31, 0100. Therefore, the twentieth century ENDED on December 31, 2000 (not 1999). January 1, 2001 begins the next century, the twenty first, which is where we are today. Ok?

Of course I know the months and the calendar hasn't been the same for all that time, but I think the point is clear.

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Essentially, it's because 0-100 AD was a century, too. So really, it's more like counting to a number but then remembering 0 was a number, and counting the numbers you counted from zero rather than 1.