I think the idea is that the muscles heat up as you run, and there is some damage done (which causes the muscles to get stronger later) and micro tears in the muscles are what cause the pain. The cold wate/ice cools the legs and reduces the inflamation, allowing the muscles to heal quicker. This is best done within a couple of hours after the race.
Residual stiffness, caused by lactic acid hanging around in the muscles, can be helped by massage, warm bath or best of all by a gentle recovery run, to flush out the toxins (if you can get your legs to move!)
If you try the cold bath (brave, very brave), let us know if it works.