Option prices encode the market's consensus on the probabilities of future moves in the underlying price. (Thus, if you speculate in options, considering some to be undervalued or overvalued, you are betting on your own probabilistic forecast that differs from the consensus.) The Black-Scholes model is based on an assumed lognormal distribution of the underlying. That is, the log return over the next time interval T is normally distributed with standard deviation IV * sqrt(T), and this implies a pricing formula for all options on that underlying in terms of a constant implied volatility (IV) parameter. Note that the option price is always an increasing function of IV (all else equal, a higher IV gives a higher option price).
Since the 1987 crash, as you note, the consensus probabilities for stock returns typically include heavier-than-lognormal tails, especially on the downside. This is reflected in higher prices for out-of-the-money puts than the Black-Scholes model predicts, because the chance of those puts becoming in the money is higher. (For example, 10 times standard deviation or "10-sigma" moves are not unheard of in the stock market, even though they would be vanishingly rare according to the normal distribution.) Thus, if we still use the Black-Scholes formula to define a per-option IV from the market prices, we find that the pricing at those downside strikes corresponds to a higher IV.
See this answer and also this question.