There are a lot of variables at play. Some of the more influential ones:

(1) The "War on Drugs" has led to mass incarceration for relatively minor offenses. Not only does this directly affect the incarceration rate, but it frequently leads to unemployment and poverty issues once released from prison.

(2) Deinstitutionalization of mental health services has driven up incarceration rates, with low risk offenders being channeled into the prison system instead of to service oriented facilities. Once again, the experience and record of incarceration generally exacerbates conditions leading to cyclical re-offending.

(3) Lack of substance recovery resources and overemphasis on punitive rather than rehabilitative response.

(4) School to prison pipeline, whereby "trouble" youth are effectively channeled into the prison industrial complex early on. Extrication from this trajectory can be very difficult.

And etc. ... such an embarrassingly broken system.

I wonder if it is to do with self-determinist attitude. In many other countries it is taken into account that there is a much stronger pull to commit crime due to psychological and social factors and that by addressing these factors you can change the person. I get the impression that in the USA there is much more a feeling that people commit crime because that is who they are - they are criminals and crime is what they do. I imagine a society that has less empathy and that this is a reason for harsher sentencing. Am I wrong?

There are many reasons behind crime rates and hence incarceration in the absence of alternative measures.

Low education, unemployment, lack of opportunity, population consolidation within welfare based low socio economic areas etc etc can all lead to the establishment of criminal activity, as well as career criminals who have no intention of a straight income.

Greece was renowned for having a low crime rate however during the past few years of austerity unemployment has risen and the country is now dealing with a crime wave.

It's not difficult to see the correlation and the commission of crime will simply rise as the jobless population grows if there is no end / future in sight.

Companies have a financial interest in more people being in jail.

Those companies lobby for, and get, "tough on crime" laws.