The simple answer to how should I store my double edge razor is any way you want. As long as you rinse any residue off the razor after shaving and dry the off sufficiently you can store it anywhere or anyway you want. Most decent razors have a plated finish that makes it resistant to corrosion or rust. You could also really kick things up a notch and pick yourself up a stainless steel razor like Feather double edge razor to offer even more protection from rust or corrosion.
If you are looking for a stand for your razor to rest on between shaves it would simply be for aesthetic reasons. There is no benefit to the razor sitting on a stand over your counter or a shelf in your cabinet. But I fall in the camp of having a nice place to hang your razor (and brush) and over the years even made myself a few stands like the ones below. If you own several razors having a good place to store them could also prove useful just to keep things tidy and organized. Personally I have a dedicated shaving cabinet with one shelf dedicated to razors.
But I would imagine most of you have a single razor. If that is the case the most important aspect of storing your razor is cleanliness. Rinsing, drying, and occasionally giving it a good scrubbing with a toothbrush and some soap with the razor disassembled. If you want to display your razor get a stand; if you are just worried about keeping your razor in working order keep it where ever you wish.