If the sole purpose of a broker is simply to buy and sell for others, why would they go bankrupt in the stock market crash? It's not like they have money invested in the cause, they just play with other people's money (or am I misunderstanding?)
To expand on Schwern's answer somewhat, using leverage as a trader works more or less like this:
Scenario 1: you have $1k of play money and use it to buy stocks. The market goes up 10%. You sell. You now have $1.1k. Net gain: $100, minus some transaction fees.
Scenario 2: you use $1k as collateral for a loan to bring your play money to $10k. The stocks go up 10%. You sell. You now have $11k. Repay the loan. Net gain: $1k, minus some financial fees and some transaction fees.
Suppose now that, in the above two scenarios, the market goes down 10% instead. The first sets you back $100; the second wipes you out.
Suppose finally that the market goes down 20% instead. In the second scenario you're not only wiped out; you also owe whoever loaned you money an extra $1k. That whoever usually is your broker, and that is how they lost money.
If they did, it would be because of the practice of buying on margin, a form of loan from your broker where you use the stocks themselves as collateral. This allows you to buy more stock than you have cash on hand. If the price goes up, you get even richer and sell some of the stock to pay off the loan. If the price of the stock falls, you'll find yourself suddenly owing a lot of money to your broker; a margin call.
This is still the norm today. The major difference is the Federal Reserve has stricter regulations on how much you can buy on margin. Now it's 50%. Then it could be up to 90%.
According to Wikipedia, brokerages were also lending a lot of money. More than could ever be repaid. The whole thing was a pyramid scheme that required the stock market value to continue to rise to support itself.
By August 1929, brokers were routinely lending small investors more than two-thirds of the face value of the stocks they were buying. Over $8.5 billion was out on loan, more than the entire amount of currency circulating in the U.S. at the time.
When the market started to fall and the brokers made margin calls people had to sell their stocks to their brokers. The more they sold the more the price dropped. With such high margins the value of the stock would quickly fall below what they owed. As stock prices fell, more margin calls would happen, prices would fall further. As the value of stocks continued to fall, less and less people would be able to repay their brokers. The brokers would have to sell their own assets to cover.
Brokers went bust due to an aspect of investing that isn't often covered by mainstream media: the margin call.
When you buy stocks on margin, meaning you're borrowing the money to buy a lot of the stocks, who puts up the rest of the money? The broker.
Buy $10k worth of stock at the 15% margin allowable in the 1920's, and you only paid $1500. So if it goes up in price, you get 100% of the increased price. Wow, isn't that great?
But, if it goes down below what you paid the 15% on, you eat 100% of the loss, and you're expected to pay the difference right now so the broker doesn't lose their equity. That's the margin call... in those days, it was literally a call from the broker you didn't want to get.
If your $10k of stock you only put up $1500 for drops to $8k, you now owe your stockbroker another $2k, and you get nothing for that $2k, just the stock you thought you owned. You do have that $2k, don't you? Or did you buy more stock on margin with that, figuring you'd cash in even more? Oops, looks like you'll have to sell off some of your other stock to make that margin call, only your other stock has just plunged in value, so you have margin calls on that, too.
This worked well as long as a lot of people were pouring money into the market and it kept going up. It doesn't work well if the stock prices drop. That tends to spook margin investors into selling everything just to limit their losses, especially if they only put up 15% to buy in but are on the hook for 100% if the price drops.
And if the market in general consists mostly of margin investors, as it did in the 1920's, even a small overall stock drop can spook a lot of people into selling, which accelerates the price drop because there aren't that many buyers...
If you wonder why some investors even today will sell stock that slips just a little, that's one reason why. They bought on a 50% margin, and want to limit their losses.
So that's what happened in 1928... prices started to drop, people started selling to cover their margin calls, no one was buying, the price really dropped, even more margin calls. You, who put up $1500 for $10k of stock that's only worth $1k now, don't have $9k, so you default on your broker.
And your broker... um... went broke because most of their clients couldn't make good on the margin calls. Add in the fact that the 1920's broker was getting their money that they loaned you by borrowing on their stocks (which were also highly inflated in value), and the whole thing became a house of cards that collapsed, because there wasn't nearly the hard currency backing the market up that everyone thought there was.
To a degree, this was mirrored in the 2007 real estate crash. Inflated prices, speculators borrowing beyond their means, and when real estate prices plummeted, the banks and investment houses that put up the mortgage expect you to pay the difference between what they loaned and what the house is now worth (it's in the mortgage agreement, in the fine print), and you don't have that money.