There are many ways to do this, but the simplest explanation is: you enter an agreement to borrow an asset and agree to give that asset back at a certain date in the future. You then immediately sell the asset. Then, on the date certain, you buy that asset back and give it back to the lender. This is called "shorting" and is very easy to do with fungible assets like publicly traded stocks, bonds, or even collateralized debt obligations. It's so easy and common that you could open up a brokerage account and the broker will actually lend you money to do this.
The problem is timing. Many, many people saw this collapse coming, and most of them got hosed by betting on it. In the above example, say that you borrowed and sold 10,000 shares of Genworth Financial stock on January 1, 2007 because you knew they were sitting on a pile of bad mortgages and the stock price would tank when the music stopped. If you agreed to give that stock back on July 1, 2007, you would have lost money, because the stock went up. If you had just made this bet a few months later, as a few people did, you would have made a killing.
Or the big boogy man from the crisis, credit default swaps. They're complex financial basically insurance that you can take out on any asset, even one you don't own. Many institutions saw the chance of these securitized mortgage assets failing as so remote that they'd sell credit default swaps for fractions of a penny on the dollar on their coverage. They were cheaper and more predictable then shorts and when the crisis hit if you held them you made a killing. Of course a lot of that killing came on the back of the U.S. taxpayer. The number of credit default swaps that they sold were what almost killed AIG, so the government stepped up and covered almost $14 billion of their losses.
The problem is timing. Many, many people saw this collapse coming, and most of them got hosed by betting on it. In the above example, say that you borrowed and sold 10,000 shares of Genworth Financial stock on January 1, 2007 because you knew they were sitting on a pile of bad mortgages and the stock price would tank when the music stopped. If you agreed to give that stock back on July 1, 2007, you would have lost money, because the stock went up. If you had just made this bet a few months later, as a few people did, you would have made a killing.