First of all, please note that there is some amount of disagreement among historians (and particularly among economic historians) as to why the boom of the 1920s could not be sustained. You will be able to find texts and scholars who disagree with the answer I will give. The answer I will give is the most common answer even though it is not universally accepted.
One reason why the boom could not be sustained was that much of the buying during the boom was on credit. This was true both of stocks and consumer goods. When people borrowed in order to buy stocks, they pushed the prices of the stocks up higher than they should have been. This bubble was doomed to burst. When people borrowed in order to buy consumer goods, they bought up all the goods they needed very quickly and the market became saturated. There were no longer as many people who needed to buy things. Many historians also say that wages were too low (and that farmers were doing very poorly) and so they could not afford to buy enough products to keep the boom going. They say that if the wealth had been more evenly spread out, people could have afforded more goods and the boom might have been more sustainable.
A second reason why the boom could not be sustained was because it was based on psychology and therefore could be undone by psychology. The boom was based, in part, on the fact that people thought that everything would continue to be great. They thought that the economy would continue to boom and so they bought many consumer goods and they invested in stocks. They did not necessarily have facts to back this up—they were simply confident. When the stock market crashed in 1929, they lost this confidence. When people lost confidence, they stopped buying. When they stopped buying, stock prices crashed. When they stopped buying consumer goods, companies laid off workers. When they laid off workers, even less buying took place and a vicious cycle ensued. Because the prosperity was based more on people’s irrational confidence than on underlying strength in the economy, the bubble was easily burst.