Because Allah commanded it: "Indeed, Allah and His angels send blessings upon the Prophet. O you who believe, send blessings upon him and greet him with peace." The phrase means "may Allah send His salāh (praise and mercy) upon him and grant him peace," and it is an act of worship, not a mere formality. The Prophet ﷺ said, "He who sends blessings upon me once, Allah sends blessings upon him ten times in return." Scholars differ on its ruling inside the prayer: the Shāfiʿī and Ḥanbalī schools hold that sending blessings upon the Prophet ﷺ in the final tashahhud is obligatory, while the Ḥanafī and Mālikī schools hold it is strongly recommended there. Outside the prayer it is emphasised whenever his name is mentioned, and especially on Friday; many scholars also advise writing and saying the phrase in full rather than abbreviating it to letters.
Q&A · General
Why do Muslims say "ṣallallāhu ʿalayhi wa sallam" after mentioning the Prophet?
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.