From Wikipedia:
"The American slang term is first recorded in 1914, the shortened form fag shortly after, in 1921.[8] Its immediate origin is unclear, but it is based on the word for "bundle of sticks", ultimately derived, via Old French, Italian and Vulgar Latin, from Latin fascis.[8][9]
The word faggot has been used in English since the late 16th century as an abusive term for women, particularly old women,[9] and reference to homosexuality may derive from this,[8][10] as female terms are often used with reference to homosexual or effeminate men (cf. nancy, sissy, queen). The application of the term to old women is possibly a shortening of the term "faggot-gatherer", applied in the 19th century to people, especially older widows, who made a meagre living by gathering and selling firewood.[10] It may also derive from the sense of "something awkward to be carried" (compare the use of the word baggage as a pejorative term for old people in general).[8]
An alternative possibility is that the word is connected with the practice of fagging in British private schools, in which younger boys performed (potentially sexual) duties for older boys although the word faggot was never used in this context, only fag. "