The bodies of Inmates on death row once being executed are sometimes claimed by family members and then the family members take the inmates body to be buried, cremated etc.
So the prison contacts the inmates family to try and have them come pickup the body or have arrangements made for the funeral home to come pickup the body of the inmate that has been executed.
If no family members are available to accept the inmates body or have anything else done to the death row inmates body then the prison has ways of disposing of the body.
Prisons usually have cemeteries where they can place the inmates body and bury the inmates body once they're executed.