I'm not sure what "get on with someone's life" means. Is this an idiom or something?

I'd consider it an idiom.

It's used when something has stopped the natural progression of things.

I usually hear it in cases where a person has become deeply involved in some project or with some person, and the deal goes bad, so to speak.

The person is emotionally devastated. The implication is that the person must now start over. And the decision to do so is made.
Okay, I understood. Thank you.