late 14c., rendren, rendre, "repeat, say again, recite; translate," from Old French rendre "give back, present, yield" (10c.) and Medieval Latin rendere, from Vulgar Latin *rendere, a variant of Latin reddere "give back, return, restore," from red- "back" (see re-) + combining form of dare "to give" (from PIE root *do- "to give").
The alteration in Vulgar Latin was perhaps simply nasalization or perhaps on analogy of its antonym, prendre "to take" (itself a contraction of prehendere). The irregular retention of -er in a French verb in English is perhaps to avoid confusion with native rend (v.) or by influence of a Middle English legalese noun render "a payment of rent," which is in part from French noun use of the infinitive.
The sense of "reduce," in reference to fats, "clarify by boiling or steaming" also is from late 14c. The meaning "hand over, yield up, deliver" is recorded from c. 1400; sense of "to return" (thanks, a verdict, etc.) is attested from late 15c., as is that of "make or cause to be) in a certain state; the meaning "represent, depict" is attested from 1590s. Related: Rendered; renderer; rendering. Also compare rendition, rent (n.1).