The reasoning has three steps.
The first step is that healing needs oxygen. As a wound heals, blood vessels open in the area so blood can bring oxygen and nutrients to the wound. Oxygen is essential for healing.
The second step is that some wounds are short of it. With ischemic ulcers, narrowed arteries prevent a healthy supply of blood from flowing to the legs. That means the tissues do not get enough nutrients and oxygen, and the lack of nutrients causes cells to die. A wound that is not healing is sometimes a wound that is not being supplied.
The third step is what the chamber does about it. The air pressure inside the chamber is about two and a half times greater than the normal pressure in the atmosphere. This pressure helps your blood carry more oxygen to organs and tissues in your body. Put more simply, it helps deliver more oxygen to the wound.
Put together: healing needs oxygen, some wounds are not getting enough, and the chamber raises how much the blood can carry.
That is not a promise of an outcome. Hyperbaric oxygen therapy can help some wounds heal faster. Which wounds, and whether yours is among them, is a clinical judgment made by people who can examine it.