An air fryer itself does not cause cancer and it's safe to cook food in an air fryer just like it's safe to cook the food in a deep fryer.
The reason that cooking in an air fryer supposedly causes cancer is that the frying of foods is supposed to create carcinogens which lead to cancer.
The air fryer or even deep fryer when cooking foods creates acrylamides, as well as polycyclic aromatic hydrocarbons and heterocyclic amines but that can result from all high heat cooking of meats.
The way that air fryers cook food without use of cooking oil is through convection.
The air fryer cooks the food by circulating hot air around the food at a high speed which cooks the food.
The air fryer has a fan that circulates the hot air from heating elements that get hot like a space heater does and that heat then cooks the food.
It creates a crisp food like deep frying does but without the food needing to be soaked in cooking oil to cook.
The air fryer uses around 1,500 watts of electricity to create that heat as well.
Basically inside the air fryer it's like an electric space heater that cooks the food.
You can also use some cooking oil spread onto the foods to help them cook better but you can just put the food in the air fryer without the cooking oil.