How does dimethylsulfoxide serve as a cryoprotectant?
This question has a simple, but non-obvious answer. DMSO and water are miscible. Mixtures, and in particular homogeneous ones like that formed by water and DMSO, usually have lower melting (or more relevant in this case, freezing) points than either substance does alone. This is the reason salt melts ice: the salt solution has a lower freezing point than pure water. The physics behind this melting point depression are complex, but a simple enough approximation involves disruption of the intermolecular forces holding the solid crystal lattice together. 

Water freezes at 0 °C, and DMSO freezes at 19 °C. Mixtures of the two have lower freezing points. More importantly, mixtures of the two likely form less disruptive crystals when frozen. Water is well know for having a solid state that is less dense than its liquid state. Ice crystals take up more room than the liquid water they formed from. Single water crystals can grow quite large, especially as smaller crystals merge. Additionally, water crystals tend to form this hexagonal fractal pattern (snowflakes) which have "pointy bits". DMSO, even at the 10% by weight customarily used, must prevent such large disruptive crystals from forming, especially when the everything is cooled to -80 °C. 

DMSO is used for this purpose instead of other water-miscible liquids such as ethanol because it has very low toxicity.