Why are deep-sea divers given a breathing mixture of helium and oxygen instead of ordinary air?

At depth, the high pressure dissolves a large amount of breathing gas in blood. Nitrogen has appreciable solubility, so on rapid ascent it comes out of solution as bubbles in the bloodstream, causing decompression sickness ('bends'). Helium has a much lower solubility (higher $K_H$) so it dissolves less under pressure and releases less rapidly on ascent.