early 15c., satisfien, "do penance," also "appease, assuage;" also "fulfill (a desire), comply with (a command), satiate (a hunger or thirst)," from Old French satisfiier "pay, repay, make reparation" (14c., Modern French satisfaire), from Latin satisfacere "discharge fully, comply with, make amends," literally "do enough."
This is from satis "enough" (from PIE root *sa- "to satisfy") + facere "to make, do, perform" (from PIE root *dhe- "to set, put").
From mid-15c. as "make amends, pay damages." The meaning "cause to have enough, supply the needs of" is by c. 1500. Of feelings, "meet or fulfill the wish, desire, or expectation of," late 15c. (Caxton). From 1510s as "assure or free from doubt or uncertainty, furnish with sufficient proof." The intransitive sense of "give satisfaction or contentment" is from c. 1600.