If someone wants simple intuition, here is what happened to the drift. It did go into the formula, believe it or not, but it came into the B-S math sort of in two ways so it cancelled out in the end.
It disappears because Black-Scholes assumes people may have different preferences for risk, but at least everyone is consistent on their own preference.
There is a drift in Black-Scholes. There needs to be some way to say how much return (or drift) you personally must get to take a certain amount of risk. More accurately, we must know how much return above the risk-free rate (or risk premium) you must get to accept one $\sigma$ of return risk.
So, imagine we had a formula that required us to somehow calibrate it for your personal tastes. How do we calibrate you, then? If you want to buy an option on stock A, we need to know how much excess return you need to suffer the risk of a stock just like stock A. If we could find another stock, B, that had identical return and risk prospects, then somehow we could "plug in" that stock's numbers to the formula for you.
But wait - why look for an identical stock? Stock A is out there and we could calibrate you with it. Stock A is the very stock to use then!
Sounds a bit fishy? Not really. Presumably, you are neither buying nor selling stock A today because you don't believe it is either overvalued or undervalued. Thus, whatever you believe in your head about the risk and return to price an option on A can use stock A as the calibration. Imagine doing just the drift term: if you think stock A is going up 15%, well, then, 15% is the right drift to use for stock A options for you. As long as you are consistent it all falls out and we don't need the drift.
This makes more sense (and this is the one of the hedges that keeps getting talked about) if you imagine that you think you have secret information that a company's stock will jump 10% later today. If you really though that you would think the stock is really, really undervalued right now, true? But, you could also buy a call and sell a put that expires tomorrow with identical strikes. That would also go up 10% later today if you are right!
If you try the example I just gave in your head, you can see that, if you are right, the call options are woefully undervalued and puts overvalued) by 10%. So, the accumulation of everyone else's preferences and beliefs - which go into both the stock and the options prices - are very different from yours. You, in fact, think that stock A has a lot more return at very low risk to occur compared to everyone else...
It doesn't spit out by giving you your own price for the option (we all share one price), but it gives a price based on a collective estimate of drift - though, oddly, we don't know what that estimate is. However, option prices would seem too low if you estimate a higher drift or lower volatility, so you would load up on options.
It definitely can fall out of the math a couple of ways, via risk-neutral pricing and Girasnov's theorem, or via a Taylor series expansion that can't ignore one of the second order terms due to quadratic variation, etc. It can also be thought of via a hedge, but that is also hard to understand. None of these is intuitive.