Roth's version of events is plausible to the extent that any such alternative history can be. For some of us, there's a reflexive belief that if something happened a certain way, it had to have happened. Or, many readers dismiss alternative histories as irrelevant simply because they contradict reality in a way that ordinary fiction doesn't—given that the scenarios in most novels, though we know they're fictional, are still possible within the real world as we know it. Yet novelistic alternative versions of reality are basically simply another branch of fiction overall, and the success of such novels with the general public proves the validity of the genre.
Alternative histories are premised on a rejection of inevitability, fate, or divine planning in human endeavor. The concept is that anything could have happened, since much is determined by chance. Still, a writer has to provide enough elements we know to be real in order to make the alternative version credible in its way.
Roth does this in The Plot Against America through his accumulation of detail. The central public figures are real-life people. Charles Lindbergh did have enormous popular support: he was a hero because of his solo flight across the Atlantic in 1927 and was the object of enormous sympathy because of the kidnapping of his baby. The anti-semitism of the time—both that of Lindbergh himself and of much of the general public—was real as well. American Nazi groups held huge rallies and combined their pro-Hitler stance with a posture of being patriotic Americans, showing huge posters of Washington, having their crowds sing the Star-Spangled Banner, and appealing to Americans' nativist prejudices.
The "America Firsters" were another group spouting conspiracy theories, many of which have continued into our own time in a slightly different guise. Even many mainstream politicians and other figures in US public life (such as the "radio priest" Father Coughlin, who had millions of listeners) asserted that there was a "Jewish plot" to get the US into World War II. Lindbergh was such a figure, and it's not inconceivable that if certain things had happened differently, he could have become president as Roth speculates.
The success of a fictional scenario depends on the author's injecting enough reality to make us ignore those elements that don't conform to what is genuine fact or is truly believable. Roth does succeed in doing this, making himself and his own real-life family the centerpiece of the story. Instead of using his fictional Zuckerman persona, for instance, he uses his own reality as the basis of the narration, making it as genuine as possible on the personal level. This, in addition to the other figures such as the real-life journalist Walter Winchell, anchors the story and provides the novel the verisimilitude it needs to succeed on its level.