I own, and regularly use, an identical Eastman Retouching Stand. A mirror directs light from underneath the negative, and, when combined with a 5x magnifying lens on a stalk, shows all the dust spots and "Efke character areas" that need SpotTone. I tried it with a kerosene lamp once, but that was hopeless. I suspect Edward did his spotting with daylight from a nearby window.
Weston made his money doing portraits, so he was probably doing a great deal of retouching, not just spotting. I'll bet he used the stand a great deal.