mid-15c., "act of granting or yielding" (especially in argumentation), from Old French concession (14c.) or directly from Latin concessionem (nominative concessio) "an allowing, conceding," noun of action from past participle stem of concedere "to give way, yield," figuratively "agree, consent, give precedence," from con-, here probably an intensive prefix (see con-), + cedere "to go, grant, give way" (from PIE root *ked- "to go, yield").
From 1610s as "the thing or point yielded." Meaning "property granted by government" is from 1650s. Sense of "grant of privilege by a government to individuals to engage in some enterprise" is from 1856, from a sense in French. Hence the meaning "grant or lease of a small part of a property for some specified purpose" (1897), the sense in concession stand "snack bar, refreshment stand."