The bedrock at Diana's Baths is Conway Granite, a coarse-grained pink-to-buff granite that intruded the region as part of the White Mountain Magma Series during Jurassic-age volcanic activity roughly 180 to 165 million years ago. The same magma series produced the surrounding peaks of the Moat Range and much of the geology of the White Mountain National Forest. The granite is hard, but it is not uniformly hard; it has joint systems, slight chemical variations, and the kind of irregular surface that running water can exploit.
The pools you see today are plunge-pool potholes. At the base of each small cascade, Lucy Brook drives a load of sand, gravel, and small cobbles in a tight circular grind, the same way you would scour a pan with a wet stone. Over thousands of years that grinding wears the granite into the rounded, smooth-walled, bathtub-shaped depressions that named the falls. Each pool drains into the next, creating the staircase profile that reads so cleanly from the wooden footbridge.
The other key piece is the Pleistocene glaciation. Until about 12,000 years ago the Laurentide Ice Sheet covered the White Mountains, scraping soil and loose rock off the granite as it advanced and retreated. When the ice finally pulled back, what was left was a freshly stripped granite surface for the post-glacial streams to start working on. Diana's Baths is a textbook example of a small-stream sculpted-pool form on a young, glacier-prepped granite surface; it is not particularly old by waterfall standards, and the shapes are still actively changing.