1530s, "act of going out," from Latin egressus "a going out," noun use of past participle of egredi "go out," from ex "out" (see ex-) + -gredi, combining form of gradi "step, go" (from PIE root *ghredh- "to walk, go"). Perhaps a back-formation from egression (early 15c.). Meaning "place of exit" is from 1670s. "One who goes out" is an egressor.

Proto-Indo-European root meaning "to walk, go." 

It forms all or part of: aggress; aggression; aggressive; centigrade; congress; degrade; degree; degression; digress; digression; egress; gradation; grade; gradual; graduate; grallatorial; gravigrade; ingredient; ingress; plantigrade; progress; progression; regress; regression; retrograde; retrogress; tardigrade; transgress; transgression.

It is the hypothetical source of/evidence for its existence is provided by: Latin gradus "a step, a pace, gait," figuratively "a step toward something, a degree of something rising by stages;" gradi "to walk, step, go;" Lithuanian gridiju, gridyti "to go, wander;" Old Church Slavonic gredo "to come;" Old Irish in-greinn "he pursues."  

late 15c., "egress," from out (adj). From 1620s, "a being out" (of something), from out (adv.). From 1764 in politics as "the party which is out of office." From 1860 in the baseball sense "act of getting an opposing player out of active play." From 1919 as "means of escape; alibi."

1550s, "a sudden rush (out), a dashing or springing forth," especially of troops, from a besieged place, attacking the besiegers, from French saillie "a rushing forth," noun use of fem. past participle of saillir "to leap," from Latin salire "to leap" (see salient (adj.)).

Hence figuratively, in 17c. of spiritual matters, in 18c. of wit, etc. In architecture, "a projection," 1660s. Sally-port "gate or passage in a fortification to afford free egress to troops in making a sally" is from 1640s (with port (n.2)).