The overhead goes to a bunch of stuff. The costs of housing inmates include the cost of hours of judges, lawyers, COs, etc. However the way it it profitable, beyond just collecting fines, and deciding your money is suspect, and taking it, is because of the corpratization of the legal system. Huge, but, often, not well known, corporations that run aspects of jails, and prisons. They make the food, run the inmate phone systems, control the inmate commissary, staff the medical departments, and more. These are just the ones that work with jails. There are third part corporations that provide bulk legal assistance work, editing services, services for a lot of the moving factors of the legal system.

These companies, in turn, give huge amounts of "donations" to politician's needs. Campaign funds being the most well known. This money, while paying for these costs, is also used to keep them living an exceptionally comfortable life. Many, after pushing through legislation favorable to a company, will be compensated in a number of ways. From them being able to take advantage of stock investment knowing how the law is about to change, and how that will affect their holdings, to exiting politics and being given a cushy, high paying, fluff job in the industry they helped out.

There isn't so much of the straight bribes, graft, and other forms of corruption people assume with politics. It is more abstracted than that, and technically legal. Obviously there are conflicts of interest that can easily be seen in this, however, since a company isn't just handing the official a bag of money, that they will keep as their own income, it is deemed legal.