Why do we say in the Tashahhud: "O Allah, send prayers upon our master Muhammad..." and does Allah send prayers upon our master Muhammad, and why do we say "peace and blessings be upon him" when mentioning the name of Muhammad?

Muslims invoke blessings upon the Prophet during their prayers in obedience to Allah's command in the Holy Quran: "Indeed, Allah and His angels send blessings upon the Prophet. O you who have believed, ask [Allah to confer] blessing upon him and ask [Allah to grant him] peace." They also invoke blessings upon him whenever his name is mentioned, for the Prophet said: "The miser is he in whose presence I am mentioned and he does not invoke blessings upon me," and "May the nose of a man be rubbed in the dust in whose presence I am mentioned and he does not invoke blessings upon me."