What President Roosevelt is saying in this quote is that we are able to affect our own destinies. We are not controlled by fate.
Many people blame their problems on fate. When they do this, Roosevelt says, they are imprisoning themselves. They become prisoners of their own minds because their minds are stopping them from achieving everything they could.
An example of this from my own life is that, for a long time, I thought I could not lose weight -- I thought my body wouldn't let me. So that's like blaming fate. But then I found out I could lose weight. That showed that when I thought I was a prisoner of fate (making me fat) I was really a prisoner of my own mind.