That's a great question. I've often been curious myself about why dogs exhibit this strange behavior of circling before going to the bathroom. I've been able to find a couple of explanations that make some sense, but I haven't been able to consult an animal behavior expert about this yet, so there may be other more reasonable explanations.

One of the explanations I read is that dogs may circle their chosen bathroom site as a way to stomp down grass and weeds in order to have a place to poop. One person suggested that originally, wild dog ancestors may have lived in areas with long or tall vegetation and they developed this instinctual behavior to maintain their hygiene (cleanliness) while defecating (pooping).

Another explanation I read is that since dogs are vulnerable to attack by predators or other dogs while squatting, they may circle in order to be able to spot any potential threats and make sure their "toilet" is safe and secure before going to the bathroom. If they find a threat, such as a predator lurking nearby, they may leave to a safer location before going to the bathroom.

Most animals (except humans) use excretory products (urine, feces) as chemical signaling. They "mark" their territory this way. Dogs sniff to find out who else has marked the territory, before leaving.

Not all dogs circle before defecating (the science word for "going #2") but many do. There are some good reasons for it. First, walking helps waste to move down the gut.

Second, dogs use body wastes to communicate with each other. Leaving urine or feces in a spot is like leaving a business card. It says who passed there, when they were there, and probably tells how they were doing. Trampling down the grass first makes the "business card" easy to see in case the other dog is upwind. Dogs also have scent glands on their feet. Often dogs scratch up the ground around their deposit, adding more scent and sight to the signal. Dogs also have glands near the anus that add more scent. (If you see a dog scooting its back end along the floor, its anal glands are probably blocked.)

So circling is probably mostly about leaving as much signal as possible. Dogs also roll in smelly things. Why do you think they do that?

Dogs are intelligent animals, just like people. And just like people, they don't like to go to the bathroom just anywhere- they usually want somewhere that is clean, where they won't be disturbed, and that is appropriate. So dogs circle to find a clean spot, or to tramp down long grasses so that they'll be undisturbed, or to make sure they are going to the bathroom in a spot free from predators (imagine walking into your bathroom and having a snake jump out at you-- dogs want to be careful). By circling and inspecting their environment, dogs make the area comfortable and safe for them to go to the bathroom.