By Groucho's normal standards, this one doesn't strike me as particularly funny, anyway. But that nstablog entry linked to by @Mehper is just about IBM's latest AI program called Watson. It doesn't say anything about where the humor comes from in this one-liner, so I'll say what I think.
I think all he means is she's frightened that if she leaves, everyone will start talking (probably negatively) about her. So by not being present, she'd become the centre of attention in a way she wouldn't like. By implication it's a boring party, where no-one has much to say about anything.
Here's a much earlier instance, from 1825,...
[He] was the life and soul of the party; bowed to everybody, danced with every lady...
...of the way "life [and soul] of the party" is more often used - someone who actively contributes to making the party go with a swing, rather than someone that everyone talks about in their absence. Groucho's one-liner turns that standard usage on its head for comic effect.