In photosynthetic tissues (such as leaves and sometimes stems), anthocyanins have been shown to act as a "sunscreen". They protect cells from high-light damage by absorbing blue-green and ultraviolet light. Ionizing radiation can damage DNA.
There are some other suggestions. The red colour of leaves may camouflage leaves from herbivores blind to red wavelengths. The pigment may signal bad taste, since anthocyanin synthesis often comes with unpalatable phenolic compounds.