This is a statement that ties into FDR's belief that individuals can summon up the individual courage needed to face incredibly challenging odds.  Given the situation in which FDR was governing with intense economic crisis situations domestically and the looming threat of European fascism abroad, it might appear that individuals might have lacked any control or any freedom of will in such situations.  Statements like these were part of the leadership quality that FDR possessed in ensuring that Americans did not feel so very helpless.  The idea of ensuring that prison is something of the mindset allowed FDR to be able to identify that no condition existed where success could not be found unless they, themselves, believed that they could not succeed.

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.

What did Franklin D. Roosevelt mean by "Men are not prisoners of fate but of their mind"?

This quote from President Franklin D. Roosevelt is from a speech that he gave in April of 1939.  He was speaking to a group of leaders of other American countries.  In April of 1939, WWII had not yet started, but there was serious trouble in Europe and Japan was already fighting in China.  It was in this context that FDR gave the quote you mention.

The meaning of this quote is that human beings have choices in everything and their lives are not dictated by fate.  Many people tend to think that they are simply unable to change various aspects of their lives.  They feel that they have just had bad luck and are doomed simply endure the consequences of this bad luck.  FDR is saying that this is not so.  He is saying that people are doomed by that sort of a passive and negative attitude.  They are “imprisoned” simply because they think they are imprisoned.

To illustrate this thought, I have heard a (probably apocryphal) story of an elephant at a circus.  This huge elephant is tied up by a rope to a small pole.  The elephant is not actually imprisoned by the rope and the pole because it could easily pull free.  However, it thinks it is securely bound and so it does not attempt to break away.  In this way, it is imprisoned only by its own mind.

FDR was saying this in the context of the problems going on in Europe at that time.  He was saying that there was nothing that was going to force European countries to start a war.  It was their own minds, and not outside circumstances, that was pushing them towards war (imprisoning them in the mindset that war was inevitable).