"Receive" has been part of English since around 1300, when it entered the language as "receiven," meaning to take something into your possession — including, notably, receiving the sacrament in a religious sense. English borrowed it from Old North French "receivre" (the Old French version was "recoivre"), which carried meanings like "seize, take hold of, welcome, accept." That French word in turn traced back to Latin "recipere," meaning "to take back, regain, recover, take to oneself, admit." Latin built this word from "re-" (back) plus a form of "capere," meaning "to take" or "to grasp" — the same root that gives us words like "capture" and "capable." The "re-" here doesn't cleanly mean "back" in an obvious way; the exact logic of combining it with "take" is a bit murky, lost in the depths of Latin word-formation.
The spelling with "ei" — rather than something more phonetic — is simply inherited from that French and Latin lineage; English didn't reshape it to look more classical the way it did with some other borrowed words, it just carried the French spelling forward largely intact. Over the centuries, the word's meaning broadened naturally: by the early 1300s it could mean "welcome someone in a certain way," soon after "catch something the way a container would," by the mid-1300s "get something as a reward," and by the late 1300s both "accept something as true" and "be on the receiving end of a blow or wound." The tech sense — receiving radio or television signals — is much newer, dating to 1908. Along the way, English also produced some now-extinct relatives from the same Latin root, like "recept" and the verb form of "receipt" ("receiten"), but only "receive" (along with "received" and "receiving") survived into everyday modern use.
receive (v.) c. 1300, receiven , "take into one's possession, accept possession of," also in reference to the sacrament, from Old North French receivre (Old French recoivre ) "seize, take hold of, pick up; welcome, accept," from Latin recipere "regain, take back, bring back, carry back, recover; take to oneself, take in, admit," from re- "back," though the exact sense here is obscure (see re- ) + -cipere , combining form of capere "to take" (from PIE root *kap- "to grasp"). From c. 1300 as "welcome (in a specified manner)." From early 14c. as "catch in the manner of a receptacle." From mid-14c. as "obtain as one's reward." From late 14c. as "accept as authoritative or true;" also late 14c. as "have a blow or wound inflicted." Radio and (later) television sense is attested from 1908. Related: Received ; receiving . Receiving line is by 1933. Other obsolete English verbs from the same Latin word in different forms included recept "to receive, take in" (early 15c., recepten , from Old French recepter , variant of receter and Latin receptus ). Also compare receipt , which also had a verb form in Middle English, receiten . also from c. 1300