The reaction mechanism of the Mitsunobu reaction is fairly complex.
Initially, the triphenyl phosphine (2) makes a nucleophilic attack upon diethyl azodicarboxylate (1) producing a betaine intermediate 3, which deprotonates the carboxylic acid (4) to form the ion pair 5. DEAD itself deprotonates the alcohol (6) forming an alkoxide that can form the key oxyphosphonium ion 8.