Faggot is apparently a bundle of sticks (formerly a burdensome baggage)

Fag in Norwegians has norse origins while faggot developed independently and only recently become a slur and come out of American vernacular.

In older English it was used to mean a burdensome person and it's sort of resumed that meaning as it's now often used to mean someone who is annoying.

Adding a fun double entendre when used to describe a homophobic person.

From fasces I think, same as fascism.