What’s the financial motivation for the state then?
For state employees (i.e. representatives, judges, etc.)? Some get kickbacks.
See: https://en.wikipedia.org/wiki/Kids_for_cash_scandal
See: https://www.nbcnews.com/news/us-news/dhs-contractors-told-wh...
Alas, the "kids for cash" scandal was such a big news event that it dominates the results for any search you'd do on the subject of private prison corruption, but it's hardly the only example.
The interests of the criminal justice system on the other hand is heavily financially benefited from the current state of affairs. More incarcerations means more judges, more lawyers, more job security.
Moves to correct this are labeled as “soft of crime” and surely my opponent Congresswoman Y isn’t pro crime?
The explosion of incarceration and the private prisons resulting from that largely come from "the war on drugs". The book The New Jim Crow is pretty good if you're interested in the topic.
"What's the financial motivation for the state" is way too blinkered and makes at least 2 different false assumptions. There are other motivations besides financial, and it doesn't have to be the state's own motivation for the state to end up doing something.
Countless government policies and programs exist which give no legitimate benefit to the state or the people, no legitimate motivation, financial or otherwise, yet they exist anyway because they do benefit and motivate someone, financially or otherwise, who has some mechanism of influence to cause it to happen.
This is almost like asking "Why would anyone do something bad?" Gosh golly why indeed? 517,000 reasons and 124 new ones every day.
i.e. its not a good motivation to increase the number of prisoners (even if one looses less money per prisoner, more prisoners will mean more loss), but it does motivate investigating ways on how one can minimize the loss on individual prisoners.