late 15c., "strap passing under the jaw of a horse" (attached to the bit of the bridle and used to restrain the animal), from Old French courbe "curb on a horse" (12c.), from Latin curvus, from curvare "to bend," from PIE root *sker- (2) "to turn, bend." The same word was used late 14c. in the sense of "a hump," and in Anglo-Latin as "curved or arched piece of timber" (late 13c.).
Meaning "enclosed framework" is from 1510s, probably originally with a notion of "curved;" extended to margins of garden beds by 1731; to "margin of joined, upright stones between a sidewalk and road" by 1791 (sometimes in this sense spelled kerb). Figurative sense of "a check, a restraint, that which holds back" is from 1610s.
1520s, of horses, "to lead to a curb," from curb (n.). Figurative sense of "bend to one's will, hold in check" is from 1580s. Related: Curbed; curbing.