Why can fuel economy be measured in square meters?

Imagine that you have a tube laid along some path and that the tube is completely filled with the fuel that you would spend to cover that path.

Area of the cross-section of that tube is the area you're asking about.

Now, if this area is bigger, the tube is thicker, which means more fuel. That is, more fuel to cover the same distance, which means lower efficiency.

Therefore, efficiency is proportional to the inverse of the area of that tube and that's why it can be measured in inverse square meters.

This reminds me of another answer about units I once posted, where I made the point that units convey some contextual information about the meaning of a number, but there is also information that is not carried by the units, and sometimes that information tells you that two quantities which are measured by the same unit are neverthelss not "compatible" in some sense (e.g. they shouldn't be added to each other). In that case, one of the examples I gave was the difference between circumference and radius. Both of these are lengths, but they mean different things, and you generally shouldn't be adding them together. It would not be the craziest thing to represent these two types of length by different units, e.g. circumference-meters and radius-meters.

Fuel economy is another one of those cases where there is extra information beyond what the standard units tell you, and it might not be the craziest thing to represent that extra information with more detailed units. Specifically: suppose you measure fuel volume in cubic meters. (A liter is, of course, 0.001 cubic meters.) Consider what those cubic meters represent. You might come to realize that it's really a product of $$\text{width-of-fuel meter}\times\text{height-of-fuel meter}\times\text{length-of-fuel meter}$$ Normally, the distinction between length, width, and height is not important, and the fact that we're talking about fuel measurements specifically is indicated by the surrounding context, so we leave those qualifiers out of the units and just say "meters". That's how you wind up reducing the unit of fuel volume to plain old cubic meters.

But in this case, when you calculate fuel economy, you wind up dividing by a completely different kind of meter: the $\text{distance-traveled meter}$. So the extra-context unit of fuel economy is $$\frac{\text{width-of-fuel meter}\times\text{height-of-fuel meter}\times\text{length-of-fuel meter}}{\text{distance-traveled meter}}$$ And in this form, it's clear that you shouldn't really be canceling width or height or length against distance traveled, just as you shouldn't be adding circumference-meters to radius-meters. Sure, all the meters are meters, but they're all measuring different things.

That's why you probably shouldn't cancel out one of the meters from the top with the meters from the bottom and leave yourself with $\mathrm{m}^2$. You can do so as a mathematical curiosity, but you've discarded some of the physical meaning in the units, and you shouldn't be too surprised that the result you get doesn't seem very physically meaningful either.

As xkcd says, the inverse fuel efficiency of a vehicle is the volume of fuel consumed per distance travelled, and this is the cross-sectional area of a pipe (or trough if it makes it easer to visualise) full of fuel down which the vehicle conceptually runs, consuming the fuel in the pipe as it goes. Obviously an inefficient vehicle will need a thicker pipe as it eats more fuel per distance travelled, and the volume of fuel it consumes is the pipe's cross-sectional area multiplied by the distance. That's why the units are $\text{length}^2$.

And efficiency is just the reciprocal of the fuel consumed per unit distance, obviously