Can someone please explain this pun?

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.

What does this phrase mean? Is it funny? ? Don’t you have to be at the party in order to be the life of it? Why would someone be afraid to be the life of a party? Is she shy? Are parties living beings? Of course, being the life of the party means that you are the center of attention. So, if you are the center of attention when not present, that means that people are talking about you. From our lived experiences, we know that gossiping – people talking about you when you are not there – is generally a bad thing. Just like Sherlock Holmes, we can deduce meaning (and humor) by making these connections.

...

Resource (and the whole article): The genius of Watson

I think the answer is in the inversion of the italics, in leaving she would improve things but doesn't want to do so. The phrase "life of the party" historically seems to refer to someone that is talked about/with while there, and in leaving would become the someone talked about who is not there, ergo becoming the death of same.

Although there's already an accepted answer. Here's an alternate interpretation I thought was worth a mention:

That by leaving she would breathe life into the party, so to speak or bring the party back to life.

It's an uncommon use of the word though because, we see death used in this context more often i.e. You'll be the death of me. Quite interesting I'd say but by Groucho's high standards, not hilarious.