In traditional Chinese culture, a person—especially men of scholar-official or noble families—often had two names:
名 (míng): the given name, usually given at birth. Here, that's Wenqing (文倾).
字 (zì): the courtesy name or style name, taken upon reaching adulthood (usually age 20 for men), meant to reflect the person's aspirations, virtues, or personality. Here, that's Muzhi (牧之).
So:
Tang Wenqing (唐文倾), styled Muzhi (字牧之) = His birth name was Tang Wenqing, and his courtesy name (used by peers, in formal settings, or after reaching adulthood) was Muzhi.
文倾 (Wénqīng): Wen means "literary" or "culture," Qing can mean "to incline" or "to lean." So possibly "one inclined toward literature or refinement."
牧之 (Mùzhī): Mu means "to herd" or "to tend," and Zhi is a classical possessive particle. "Muzhi" has a literary, poetic flavor—like "He who tends" or "the one who shepherds."
This kind of naming was very common in imperial China, especially among scholars and nobility. Many historical figures are known more by their courtesy names (字) than their given names.