Faggot
Middle English (in the sense ‘bundle of sticks for fuel’): from Old French fagot, from Italian fagotto, based on Greek phakelos ‘bundle’.
wiktionary
From Middle English fagot, from Middle French fagot(“bundle of sticks”) in turn from the Medieval Latin and Italian fagotto. Compare Old Occitan fagot, Italian fagotto, fangotto, Spanish fajo(“bundle, wad”). Perhaps from a diminutive of Vulgar Latin *facus, from Latin fascis(“bundle of wood”). The senses relating to persons, though possibly originating as an extension of the sense "bundle of sticks", could have been reinforced by Yiddish פֿייגעלע (feygele). Doublet of fagotto. See also: fag.
etymonline
faggot (n.1)
late 13c., "bundle of twigs bound up," also fagald, faggald, from Old French fagot "bundle of sticks" (13c.), of uncertain origin, probably from Italian fagotto "bundle of sticks," diminutive of Vulgar Latin *facus, from Latin fascis "bundle of wood" (see fasces). But another theory traces the Vulgar Latin word to Greek phakelos "bundle," which is probably Pre-Greek.
Especially used for burning heretics (emblematic of this from 1550s), so that phrase fire and faggot was used to indicate "punishment of a heretic." Heretics who recanted were required to wear an embroidered figure of a faggot on the sleeve as an emblem and reminder of what they deserved.
Faggots, the traditional British dish made from the innards of pigs (liver, lungs, heart, spleen) mixed with bread crumbs, rolled in balls, and braised in stock (1851) apparently is the same word, presumably from the notion of "little bits and pieces bound up together."
faggot (n.2)
"male homosexual," 1914, American English slang, probably from earlier contemptuous term for "woman" (1590s), especially an old and unpleasant one, in reference to faggot (n.1) "bundle of sticks," as something awkward that has to be carried (compare baggage "worthless woman," 1590s). It may also be reinforced by Yiddish faygele "homosexual" (n.), literally "little bird." It also may have roots in British public school slang noun fag "a junior who does certain duties for a senior" (1785), with suggestions of "catamite," from fag (v.). This also spun off a verb (see fag (v.2).
He [the prefect] used to fag me to blow the chapel organ for him. ["Boy's Own Paper," Aug. 3, 1889]
Other obsolete British senses of faggot were "man hired into military service merely to fill out the ranks at muster" (1700) and "vote manufactured for party purposes" (1817).
The explanation that male homosexuals were called faggots because they were burned at the stake as punishment is an etymological urban legend. Burning sometimes was a punishment meted out to homosexuals in Christian Europe (on the suggestion of the Biblical fate of Sodom and Gomorrah), but in England, where parliament had made homosexuality a capital offense in 1533, hanging was the method prescribed. Use of faggot in connection with public executions had long been obscure English historical trivia by the time the word began to be used for "male homosexual" in 20th century American slang, whereas the contemptuous slang word for "woman" (in common with the other possible sources or influences listed here) was in active use early 20c., by D.H. Lawrence and James Joyce, among others.
