The word fag, or faggot, is the 4chan general insult for anything that Anonymous does not like, including himself. Contrary to popular belief, this meaning is exclusive to 4chan and does not necessarily imply a disdain for a non-mainstream sexual preference, though it can given the context.
By extension, faggotry is the science of acting like a “faggot”. Fag is frequently affixed to common words and memes to give them a negative connotation, though by 4chan standards it is only a mild insult when not paired up with other curse words. For example, a Namefag is somebody using a name, while a fagfag is a smoker, a straightfag is straight and a homofag is a homo.