Assume that you want to buy an option on Apple stock - 3 months in the future. Assume that price today is $500. If you want to buy an option with strike=500, you will pay 50 vol (BS as a formula will just convert the price in vol terms into a dollar cash price). But - if you want to buy an option struck at 400 - you might pay 75 vol. And if you want a 300 strike - it will be 100 vol.
As you see, volatility will RISE the further you move away from spot price (ie where it is today). So while BS calculation assumes constant volatility to convert vol to dollars, traders always ask for higher vol, as you move out to less likely scenarios. That is how they deal with the world being non-normal. It really is options 101 - so any writer who talks about BS assuming Gaussian probabilities is either lying or doesn't know anything about the subject.