I think the idea was that computers wouldn't be able to calculate making their clocks switch from 1999 to 2000, so all the unmanned systems would crash and power plants would explode or some such.
But they didn't.
Basically there was almost no software written at the time that would handle a year greater than 1999, due to lack of foresight. (Years were counted with two digits in most software, meaning "00" would look like 1900.)
Everything had to be be updated really fast because just like with IPv6 everyone waited until the last minute. A lot of things did get hit by the Y2K bug, but nothing as big as what the scare stories were selling.
The next such thing we have to worry about is the 2038 bug.
In retrospect, the idea of trains being derailed and planes falling from the sky was silly.
That was mostly the loonies saying that, but I know other people filled their bathtubs with water and prepared for the apocalypse anyways.