How can I improve the texture of the skin on my hands? Last year my hands suddenly appeared very crepey on the surface. (photo)

With regards to your specific question, aging of the hands like all other aging is a function of a number of factors including genetics and lifestyle (weight changes, sun exposure, occupational exposure). While there is no definitive answer for "why," there are options.Aging of the hands manifest (aesthetically speaking) in skin changes and volume loss. Skin changes may take the form of pigment variations, fine lines etc. Volume loss leads to prominence of the tendons and veins on the dorsum of the hand.
I typically treat the skin with Fraxel laser therapy.
The mainstays of volume loss correction are fillers. HA's (Juvederm and Restylane) and Radiesse (hydroxyapatite) are the most commonly employed. I prefer HA's for a smoother contour relative to hydroxyapatite, however, compared Radiesse, the effect is shorter lived.
I will typically dilute my HA in 1 cc of local anesthetic for a total of 2 cc per hand.
Both should be infiltrated superficially. I prefer blunt cannulas to avoid bruising and disruption of dorsal veins. After injection, the product should be aggressively massaged to a smooth contour.As always, discuss your concerns with a board certified plastic surgeon (ABPS)

From your picture your hands have enough volume so that any type of volume increase is not a good idea. Besides keeping your hands out of the sun, moisturizing, not smoking etc. the best options are micro needling with PRP several times with weeks apart from each treatment. An expensive way of dealing with this would be stem cell isolation from fat and placing the stem cells evenly under the skin. Different lasers can also help. Good luck