You're basically right as far as I know. Most animal cells contain water and are encapsulated by only a thin cell membrane. When the water freezes, it expands and crystalizes. This causes the cell to burst. Rigid cell walls like the ones found in plant cells or bacteria can potentially help the cell survive as can "antifreeze" molecules dissolved in the cellular fluid as mentioned above (think of the way salt water can get below freezing temps without turning to ice). Some creatures can also dehydrate themselves to keep from freezing like the microscopic waterbear.