How did the stock market cash trigger a chain of events that led to the depression?

The stock market crashed on the day in history known as Black Tuesday, October 29, 1929. Billions of dollars were wiped out in just hours. Many speculators who had bought stock on margin lost everything they had. The banks were one of the institutions that felt the effects of the crash first. In the start of the crisis, people who had put their money into banks were withdrawing it all. They feared that all of their money would be lost so they withdrew it. Unfortunately for them, the banks didn't have the money. Banks take risks by investing the money that people deposit, hoping to make money from it. When the market crashed, the banks lost all of their investments and could not afford to give everyone their money back so they had no choice but to shut down. 
Then businesses quickly, one by one, close. The businesses lost all of their money in the crash and could not afford to pay all of their workers. Unemployment quickly rose. By 1933, nearly 25% of all Americans had lost their jobs. The depression eventually went global, sending nearly every country into a state of frenzy. It wouldn't be until World War 2 that things started to get better. 

Now I know your question states what lead to the depression, but there is no definite cause that lead to it. Even today, historians and economists disagree on the exact causes. Some say that it was the result of a contraction in the money supply, others say it was the lack of government interference in the economy. But what they can be sure of, is that even before 1929, Europe and rural America had economic hardships, along with the uneven distribution of wealth, and over speculation in the stock market. Combined it created dangerous economic conditions, leading to the Great Depression.

Aor b but i'm going with b