The Black-Scholes model provides a recipe for creating synthetic options that will (under the assumption of known volatility) act just like real ones. Which is convenient because you can create synthetic versions of options that people want to have but which are not traded. Leading up to the '87 market crash, lots and lots of these synthetic options were created. Then came the crash.
People can debate endlessly about why the crash started. But once it did, there is no question that trading algorithms attempted to close out trades that were necessary to maintain synthetic options. These large trades attempted to execute in markets that had seized up, and made the market much, much worse. The result contributed greatly to the crash, and caused the synthetic options to fail to work as promised. (Besides, the Black-Scholes algorithm guarantees that it acts like an option through a certain amount of variation in the stock price, and not for a particular time period. The volatility of the crash demonstrated the importance of this discrepancy.)
Now do you see how the Black-Scholes model contributed to that crash?