When Jewishness loses its public personna and becomes a strictly private affair it takes on the character of an obsession. To define oneself in terms of one's Jewishness is to define Jewishness as an essential characteristic of personality. Down that path lies the danger of racism.
The disease of antisemitism is linked to the absence of a public, ideological definition of what it means to be Jewish and why Jews should be Jews. Jews need to be clear about their mission or they could become subject to the most vicious, violent assaults. To be Jewish should define what one does rather than what one is. With a national mission, it becomes reasonable to tie one's actions to one's ethnic identity in a way that is virtually impossible in the absence of such a mission.
Arendt's argument comes down to the notion that the assimilated Jews of Western Europe had become identified in their Jewishness rather than in their actions. Because they had already lost their spiritual identity as Jews when they divested themselves of their ritual practice the Jewishness had become a hollowed out husk, leaving little to be done but to play the role of the exotic.