Brief Summary
The Set-Up
President Roosevelt knew he was taking on a huge task in leading the nation during the Great Depression, and he had to get the public on his side—but quick.
The speech "First Fireside Chat," given just a few days after he took office, was his way of introducing himself to America, building camaraderie, and laying out his plan to get the country fixed.
The Text
Think of "First Fireside Chat" as less of a speech, and more of a, well, chat. By the side of a fire.
In a world with no television, President Roosevelt was coming as close as he could to the American people by talking through the radio on a Sunday night. Radios were a common household item by 1933—even though many people were poor and struggling—and they were often kept in the living room.
Therefore FDR's speech had all the right pieces in place for sending a personal, calming message. You've got the family together in the living room, perhaps a fire is going, it's the end of a family-oriented weekend day, and your president is giving you a little chit-chat.
You might not think that Econ 101 constitutes a "little chat," but at the time people were hungry for news and knowledge about what was happening to their country (and more importantly to their money). Also, dude had a very calming voice.
FDR spends most of this fourteen-minute speech discussing banks—how they work, why he proposed a bank holiday (translation: shutting the banks down), his plan to get them up and running again, and why the people should trust him and the government.
Roosevelt uses a lot of key words to assure the American people that he's in this fight with 'em, and that he and the government are doing everything they can to right the situation. He focuses on calming and reassuring the public, which was an essential first step to restoring order.
TL;DR
Keep calm and keep your money in the bank, y'all.