It’s the notion that you want to start the new year with your special someone. The kiss used to denote rejoicing in their relationship and starting it’s “new year” the right way, by being happy and intimate. That was back then. Now it has turned into “must kiss someone at midnight otherwise I’ll look lonely and uncool.”
Wikianswers was unable to come up with a better answer than what I said.
Though, this article says that according to English/German legend “the first person you came in contact with when the bells chimed twelve, be it a familiar, friendly face or inauspicious acquaintance, set the tone for your happiness and fortune in the coming year. A kiss with one you loved ensured affections and good tidings. To smooch someone less favorable, or not at all, could mean misfortune.”
Interestingly, kissing under the mistletoe is from an old legend where it’s said that if a girl receives no kisses under the mistletoe around Xmas time, then she will not get married in the new year. source