In terms of the cellular and molecular level, many of the relatively weak interactions holding a person together are disrupted by cold temperatures. As a cell freezes, most of its proteins and lipid membrane content will denature and "fall apart", the same as if you boiled it.
As with many things in biology, the problem is water. At room temperature, the lipid bilayers and proteins of a cell are held together largely by way of a complex network of interactions with the surrounding water. If the cell gets too hot or too cold this network of interactions between biomolecules and water falls apart. This leads to a disruption of the cell's overall organization*, which in turn can cause problems up to and including complete cell rupture and death.
So in order to preserve a living creature under cryonic conditions you need to replace the water in and around its cells with a different molecule that will maintain most of the normal intermolecular interactions even under conditions of extreme cold. There are protocols for freezing cell cultures that use various different substances as replacements for water, such as glycerol or DMSO, and there's a frog that loads its own blood with carbohydrates at low temperatures in order to turn it into a better cryonic medium.
*The disruption of a cell's organization at extreme cold is likely to persist even if the cell is brought back to room temperature. See Gulevsky, et al for a complete review of the thermodynamics of cold denaturation.