Kinda like phosgene gas.
Upon dispersion with high explosives, the White Phosphorus rapidly reacts with oxygen in the following reaction:
P(elemental) -> P2O5 + other phosphate oxides
The military use of this derives from the rapid liberation of these oxides. It forms a cloud of "smoke", which are actually these phosphate oxides dispersed throughout the air.
These oxides are poisonous to humans all by themselves. However, these oxides are not what cause the horrible disfiguration of humans. After these oxides land on human skin, the following reaction occurs:
P2O5 + H20 -> H3PO4
The water in human skin converts the phosphate oxide to the highly corrosive Phosphoric Acid. This acid begins to burn human skin, lungs and eyes IMMEDIATELY upon contact. Severe exposure to Phosphoric acid by inhalation causes chemical pneumonitis (chemical induced pneumonia), much like Phosgene gas used in World War I. Skin contact results in the now familiar burns and leathery skin.