late 14c., "alleviate (pain, etc.), mitigate; afford comfort; allow respite; diminish the pressure of," also "give alms to, provide for;" also figuratively, "take heart, cheer up;" from Old French relever "to raise, relieve" (11c.) and directly from Latin relevare "to raise, alleviate, lift up, free from a burden," from re-, intensive prefix (see re-), + levare "to lift up, lighten," from levis "not heavy" (from PIE root *legwh- "not heavy, having little weight").
The notion is "to raise (someone) out of trouble." From c. 1400 as "advance to the rescue in battle;" also "return from battle; recall (troops)." Meaning "release from duty" is from early 15c. Related: relieved; relieving.