Since you mentioned being interested in a geometric proof, here's one that should be easy to understand (albeit with one sneaky detail swept under the rug). Start with a $w\times h$ rectangle, with $w\gt h$; we'll prove that there's another rectangle of the same perimiter but greater area. Since the area of the rectangle is twice the area of the triangle of base $w$ and height $h$, we can just consider the triangle's area. But now consider adding some small amount $x$ to the height $h$ and subtracting the same amount from the width $w$. This doesn't change the perimeter, since we're just redistributing a small segment, but we can see what it does to the area:
The area of the original triangle with base $w$ and height $h$ is the sum of the pink and green triangles, while the area of the new triangle with base $w-x$ and height $h+x$ is the sum of the pink and blue triangles. But as long as $w-x\gt h+x$, the blue triangle will have a greater area than the green one: they both have the same base ($x$) and the blue one has a greater height. This implies that as long as $w\gt h$, we can increase the area of the triangle (and thus the rectangle) by trading off some amount of width for the same amount of height, and that in turn implies that the maximum area must be achieved by the square.