As distinct from the above writeup, the phrase "speed costs money" can also be interpreted to mean the massive engine power increase needed to sustain an only slightly higher speed.
The two main retarding forces acting on a non-accelerating car are rolling resistance from the tires, and air drag. Rolling resistance is based on the tire and the car's weight, while air drag is based on the car's size and shape.
The power required to overcome rolling resistance increases linearly with speed, so if it takes 5 horsepower to overcome rolling resistance at 30 MPH, it will take 10 horsepower to overcome rolling resistance at 60 MPH. Therefore, fuel mileage would be more or less the same no matter what speed you drive at, because even though you are consuming twice as much fuel per unit time at 60, you will cover twice as much distance in the same time. In actuality, you would get BETTER gas mileage by driving faster because gasoline-powered Otto Cycle engines have increased efficiency with increased load.
However, in the real world, air drag exists as well. The power required to overcome air drag increases with the CUBE of the speed. So if it takes 5 horsepower to overcome air drag at 30 MPH, it will take 45 horsepower to overcome air drag at 60 MPH! Even when you take into account the fact that you will cover more distance by travelling at 60, this is only a linear quantity being subtracted from a cubic quantity. A square quantity will still remain.
Interpretation of the above would suggest that fuel mileage is inversely proportional to the square of the speed. This would be ignoring rolling resistance, and any efficiency increase caused by the higher load. However, those two factors are usually trivial compared to the massive effect of air drag, so it is usually okay to ignore them.
So, in conclusion, speed does cost money. Money in an amount proportional to the square of the speed.