Why can it hurt your hands when you carry heavy groceries in carrier bags?

There are a number of ways that carrying heavy bags can hurt or damage your hands, including sprain and strain injuries to the muscles, ligaments, and tendons of the hands and wrists. In addition, if one has arthritis in the finger or hand joints, it does not require much squeezing or pressure to cause pain.
Carrying grocery bags can cause pain in the absence of arthritis. The reason is that the pressure of the bag handles is quite high and concentrated over a small area of soft tissue in the inner hand. Pressure is defined as force per unit area. Pressure increases when the force increases. Pressure also increases when the area supporting the force gets very small. In this case, the force is the weight of the groceries in the bags, and the area is the region of contact between the grocery bags and your hands. You specified heavy grocery bags, so it is a significant force. This is especially true of disposable plastic shopping bags, in which the handles become very narrow when stretched and the contact area with your hands is very small. This increases the pressure substantially, effectively concentrating all of the force of gravity acting on your heavy groceries into a very small area.
Further, since the pressure is applied across the insides of your fingers, it can cut off blood flow, especially in the veins, causing blood to be trapped in your fingers. This is similar to when your foot "goes to sleep," and it is known to be painful.
One way to reduce the pain would be to carry your groceries in tote bags having wide straps. Switching to tote bags having a handle width of 1" would reduce the pressure on your fingers significantly. Another option, of course, is to make more trips carrying fewer bags. Alternatively, you could get a cart of some kind so that you are not supporting the weight of the bags at all.