The closest definition is in TLFi Se suffire [En parlant d’une pers.]: Se réaliser, satisfaire ses aspirations, sans le secours d’autre chose ou sans le secours d’autrui.

“To achieve personal success, to live up to one’s expectations, without help from something else, or without help from someone else” (my translation).

It could be translated as “make it on one’s own”.

In theory you could remove “à lui même” (since it is redundant), but the idiom is generally expressed in this way; so it’s a guarantee that you are going to be understood. The repetition is merely an emphasis (like “make it on one’s very own”; but there is no need to translate it in that way).

According to Larousse on-line “se suffire à soi-même“ is an established expression [redundant or not] meaning “n’avoir besoin d’aucun complément, d’aucune amelioration,” and the example of its use given in my personal Le Robert-Micro is very similar to the one used by “Larousse*:
“Cette definition se suffit à elle-même.”

I’d probably agree that adding “à soi-même” for emphasis to “se suffire” when used to describe [the notion of] people as being “self-sufficient” might qualify as a redundancy (perhaps in recognition of this my Le Robert has “à lui-même” in parentheses in its example of “se suffire” when used with “personnes”: “Le pays se suffit (à lui-même) … pour l’energie”).
However, when the full expression is used with nouns for “things” (such as “syndicalisme”), the potentially-redundancy-causing notion of “being self-sufficient” is, in my mind, replaced by that of “being sufficient, in and of itself” or “being sufficient, [all alone] by itself.”

Granted, if your text had talked of “le syndicat” (which is comprised of people, just as “le pays” is in the Le Robert example) as being “self-sufficient” [to accomplish anarchy] instead of the thing/idea of “le syndicalisme” as being “sufficient, by itself” [to be anarchy], then I think there would have been a slightly better case for putting “à lui-même” in parentheses (à la Le Robert) or even for omitting it altogether.