Learn English – What do you call feeling what someone else feels

I know you're looking for a feeling, in either single noun or adjective terms, and I believe that we could say that the audience feels a sense of...

justice.

However, I think there are a few other common expressions that might also satisfy your general question:

The audience felt that...

... he got his comeuppance.

... justice had been served.

... he got his just deserts.

All of these convey the idea that someone who should be punished (e.g. the villain) did indeed get punished.

I believe merely uttering these expressions conveys the sense of satisfaction that you are asking about.

The act of reflecting on one's perceived woes is also known as self-commiserating.

And then he sighed a querulous, self-commiserating sigh, as if in pure regret that he, the loved and courted of so many worshippers, should be now abandoned to the mercy of a harsh, exacting, cold-hearted woman like that, and even glad of what kindness she chose to bestow, (Chapter 48, The Tenant of Wildfell Hall, Anne Bronte)