The primary impact that Roosevelt had in terms of the public was to bring them into a closer fold into how government works. Instead of allowing them to feel shame about unemployment or business failure, which was something that was experienced prior to FDR, he convinced them that government could be part of the solution and that the crisis might have been minimized with a greater and more expansive role of government. Public works projects were driven to provide more employment to people. Legislation was created to protect consumers. The Fireside Chat was a way to bring more people in to the political process. Overall, the idea of bringing back the common touch to politics and the common man into political legislation was one of the strongest impacts FDR had on the office.
The main impact that FDR had on the people was to give them some amount of hope. The Great Depression had caused many Americans to lose hope and it was FDR's attitude and his new programs that led them to be more hopeful.
FDR's programs did not end the Depression, though. They made things better, but did not actually end it. The Depression would not end until the US started gearing up for WWII.
So FDR's major impact was to make people less pessimistic about their lives and about the nation's prospects.