It is widely known that freezing breaks soft, wet materials. However, the mechanism underlying this damage is still not clear. To understand this process, we freeze model, brittle hydrogel samples, while observing the growth of ice-filled cracks that break these apart. We show that damage is not caused by the expansion of water upon freezing or the growth of ice-filled cavities in the hydrogel that exert pressure on the surrounding material. Instead, local ice growth dehydrates the adjacent hydrogel, leading to drying-induced fracture. This dehydration is driven by the process of cryosuction, whereby undercooled ice sucks nearby water toward itself, feeding ice growth. Our results highlight the strong analogy between freezing damage and desiccation cracking, which we anticipate being useful for developing an understanding of both topics. Our results should also give useful insights into a wide range of freezing processes, including cryopreservation, food science, and frost heave.