Visualizing -- or drawing -- a product of sets as a rectangular grid where the rows are labelled by one set and the columns by the other set helps a lot, I think.
Pick a subset of the rows and a subset of the columns, and look at the points that are in both subsets. That's a product of subsets. It always looks kind of rectangular, though possibly with some rows and columns removed from a complete rectangle.
Pick some random collection of points in your original grid. Does this collection of points look like a product of subsets? You should be able to find some collections that don't look rectangular at all. They can't be products of subsets.
I think this visualization is important, but the tricky part is making this reasoning into a precise argument.
One way to do it is to notice that in a product $A \times B$, if you pick $a \in A$ and look at all the elements of $B$ it's paired with, they're always the same ones, no matter which $a$ you pick.
In particular, if you have a set of pairs and $(a,x)$ is in your set and $(b,y)$ is in your set, then for it to be a product of subsets you know the elements that $a$ and $b$ are paired with need to be the same, so $(a,y)$ and $(b,x)$ need to be in there too. (But note that $(a,b)$ doesn't need to be.)
In the visualization, this means if a set of pairs has two opposite corners of a rectangle in it, then in order for it to be a product of subsets it must also have the other two corners of the rectangle. So subsets which contain fragments of rectangles with missing corners can't be written as a product.