faggot (n.1) Look up faggot at Dictionary.com
late 13c., "bundle of twigs bound up," from Old French fagot "bundle of sticks" (13c.), of uncertain origin, probably from Italian faggotto, diminutive of Vulgar Latin *facus, from Latin fascis "bundle of wood" (see fasces).

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 their sleeve, as an emblem and reminder of what they deserved.
faggot (n.2) Look up faggot at Dictionary.com
"male homosexual," 1914, American English slang (shortened form fag is from 1921), 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," literally "little bird." It also may have roots in British public school slang fag "a junior who does certain duties for a senior" (1785), with suggestions of "catamite," from fag (v.). This also was used as a verb.
He [the prefect] used to fag me to blow the chapel organ for him. ["Boy's Own Paper," 1889]
Other obsolete senses of faggot were "man hired into military service simply to fill out the ranks at muster" (1700) and "vote manufactured for party purposes" (1817).

The oft-reprinted assertion that male homosexuals were called faggots because they were burned at the stake as punishment is an etymological urban legend. Burning was sometimes 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. Any use of faggot in connection with public executions had long become an English historical obscurity by the time the word began to be used for "male homosexual" in 20th century American slang, whereas the contemptuous slang word for "woman" (and the other possible sources or influences listed here) was in active use. It was used in this sense in early 20c. by D.H. Lawrence and James Joyce, among others.