Legitimacy is how credible your claim of being an American successor state looks. Radical departures from old American ideals obviously tarnish your image as a successor. As a Revivalist movement rather than an established faction, you begin with -2 Legitimacy (it'd be worse if you were Resistance, because hoo boy). In general, Legitimacy is something I compare between factions; it comes into play when people are trying to decide what they think of you.
The faction with the highest Legitimacy (currently New York) is the benchmark; anybody short of that is viewed as meaningfully less legitimate as a successor of the old USA. This, incidentally, is why New York is the favorite charity case for old American allies but gets nothing from the rest of the world; old allies liked how things were when America had a pulse, and they want to keep the thing that most resembles it around. In case of future revival, you know? Meanwhile, the rest of the world doesn't remember America fondly enough to care to invest in a city squatting under a sword of Damocles.
That said, Legitimacy is most useful to a strict Revivalist faction. If you actually want to build something intensely different from the old system, low Legitimacy is beneficial to you over the long term, assuming you go about this remotely honestly. Low Legitimacy means that people let go of some of their preconceptions. You clearly aren't a USA successor, so nobody expects you to act like one. While it also means some will be inclined to see you as illegitimate usurpers, it clears your plate of having to play a part you may not necessarily want to play.
Neutral Legitimacy, which in practice is, "Less than the Free City of New York's, but not critically so," means that, as a faction, you really just fail to stand out. You're another faction emerging out of the chaos. Neither the old Country born anew nor some radical, (potential) dark mirror of Victoria.
I wouldn't recommend trying to game this too hard; Legitimacy shifts constantly as other factions make decisions, and given the nature of the system, the value of a single point of the stuff won't stay stable. It's meant to quantify your polity's RP comparative to others', not serve as a super-crunchy stat. That said, at game start, FCNY leads the rankings with Legitimacy 9, and Victoria is the bottom finalist with Legitimacy -12.
As far as Legitimacy has mechanical effects, it happens at higher levels. Any faction that hits Legitimacy 25 gains international recognition as the rightful successor to the United States of America, gaining free casus bellis on all territory once held by the former United States. Low Legitimacy does not have a similar threshold, but again, it's better if you intend to build something different.
