mid-13c., avauncen (transitive), "improve (something), further the development of," from Old French avancir, avancier "move forward, go forward, set forward" (12c., Modern French avancer), from Vulgar Latin *abanteare (source of Italian avanzare, Spanish avanzar). This is from Late Latin abante "from before," composed of ab "from" (see ab-) + ante "before, in front of, against" (from PIE root *ant- "front, forehead").
Compare French avant "before" (as in avant-garde), which is from the same Late Latin word. The unetymological -d- in English was inserted 16c. on the mistaken notion that initial syllable was from Latin ad-.
It is attested from c. 1300 as "to promote, raise to a higher rank." The intransitive sense of "move forward, move further in front" is by mid-14c.; the transitive meaning "bring forward in place, move (something) forward" is from c. 1500. The meaning "give (money, etc.) before it is legally due" is attested by 1670s. Related: Advanced; advancing. The adjective (in advance warning, etc.) is recorded from 1843.