At least part of the answer is that some animals evolved [ice-binding proteins or antifreeze proteins](https://www.newscientist.com/article/mg21028083-300-polar-animals-antifreeze-has-a-spiky-secret/). These proteins prevent the growth of ice crystals which can rupture cells.
The main danger with freezing is that the ice crystals that form poke holes in the cell membranes. Without functional cell membranes there is nothing (or at least not enough) to keep your cell-stuff in place and the cell dies pretty quickly. This is also how all antibacterial _chemicals_ like soaps and alcohol work (NOT antibiotics).
Antifreeze prevents the formation of these ice crystals, or at least keeps them limited in some way that they do less or no damage to the cell membranes. This is actually used in microbiological labs; if you want to freeze your cells or bacteria, you put them with a bit of some specific antifreeze component.
Unfortunately for most cells and bacteria antifreeze is also very toxic. But, some animals have evolved components that act with a similar function as antifreeze.
We don't know! I'm working in a lab right now studying insects that can survive being frozen. They have certain characteristics that can control ice crystal growth which probably helps.