What does area represent? Since any two Euclidean shapes have an infinite number of points inside of them, shapes with different area have the same infinite number of points in them (and any object has the same number of points in it as are inside the plane it is inside). So area isn't a measure of the amount of points in an object, right? And if this is true, what does the area of a shape actually represent? In other words beyond the abstract idea of the amount of space in an object, what does an area of, for example, $4$, mean?

It tells you that you could cover the region with 4 $1\times 1$ squares. If it's a weird shape, it would be more informative to say that if you cut it out of material and a $1\times 1$ square of that material weighs 1 ounce, then the shape you cut out would weigh 4 ounces.
