If the prisoner is sentenced to LWOP or Life Without Parole then yes the prisoner will actually die in prison.
Life without parole means that the prisoner will remain inside the prison until they die of health related issues or die of old age.
So if you're sentenced life without parole and you live to 110 then you'll remain in prison until you reach that age and die.
Most prisoners die way before they reach 110 though but if you go to prison for life then that means the rest of your life.
Being in prison for the rest of your life is like being buried alive and you're basically dead inside and not really living even though your physical body is alive.
I could not imagine waking up everyday in a prison cell and knowing that it will be the same everyday and I can never leave until I die.