The Wikipedia article says:
Suppose that there is such a dissection. Make a face of C its horizontal base. The base is divided into a perfect squared rectangle R by the cubes which rest on it. Each corner square of R has a smaller adjacent edge square, and R's smallest edge square is adjacent to smaller squares not on the edge. Therefore, the smallest square s1 in R is surrounded by larger, and therefore higher, cubes on all four sides. Hence the upper face of the cube on s1 is divided into a perfect squared square by the cubes which rest on it. Let s2 be the smallest square in this dissection. The sequence of squares s1, s2, ... is infinite and the corresponding cubes are infinite in number. This contradicts our original supposition.
Why is it important that the smallest rectangle is not located on the edge?
The argument in Wikipedia is not explained well. The point is that the smallest square, $S$ say, could lie on an edge or a corner, but, whatever happens, it has strictly larger squares alongside the $2$, $3$ or $4$ of its edges that are internal to the face $C$. The cubes on top of those larger squares will build walls around the space on top of $S$ that prevent us filling the space on top of the cube over $S$ with cubes that are larger than the cube over $S$.