What does it mean to be going 40 mph (or 64 kph, etc.) at a given moment?

I think there is a very clear meaning in the physical world: If, at some moment, you were going 40 mph, if you were to stop de/accelerating and just hold that velocity, you would cover 40 miles in 1 hour.

If you were to try to measure instantaneous speed as you described, you would in fact have traveled 0 miles in 0 time and 0/0 is undefined If, however, you look at your average speed over smaller and smaller periods of time around the instant you care about--that is, (distance traveled from $t=t_0$ to $t=t_0+\epsilon$ for various small values of ε--and these average speeds "converge" (they all get closer to a single value as ε gets closer to 0), then we say that the instantaneous speed is that single value upon which the average speeds around that point converge. This is, of course, a somewhat informal explanation; to be more precise requires getting into differential calculus.

This is similar to other answers. Imagine another car beside yours. That car is covering a distance of 40 miles over the next hour at a constant velocity. At the point at which you keep pace with that car (relative velocity = 0), you are traveling at 40 mph.