This 6.5 centimeters, while relatively small now, is continuing to grow at an increasing rate, as the ice covering Antarctica and Greenland is being melted away by a higher global temperature. All this ice runs into the ocean, increasing the total volume of water in our oceans. Another significant factor in rising sea level is Thermal Expansion. The same heat that is melting the ice caps, is also heating the mean temperature of the ocean. This causes the individual molecules of water to move more vigorously, thus taking up more space. The video below better explains the combination of the two phenomena and their impacts of sea levels.
As far as the movie, The Day After Tomorrow, is concerned, while the basic premises of thermodynamics are correct and the movie does a good job of explaining the science, the time it takes for the entire process to happen in the movie is way too quick. Even if some idiot scientist cracked half of a glacier and caused it into the sea, the effects of such an event would speed up the process, but not enough for a global winter in 15 days.