This claim is a common misconception that remains false. Cracking one's knuckles has no correlation to the development of arthritis.
Joints, including knuckles, are surrounded and oiled by a fluid called synovial fluid. This fluid helps protect the ends of bones and reduces friction during movement. When cracking your knuckles, it causes the joints to pull apart and create a gas bubble. The popping noise created by the action of cracking knuckles are the sounds of gas bubbles bursting in the joint. Though this sound may be alarming, there is no extreme harm to cracking knuckles other than tentative soreness.
However, if an individual is already suffering from osteoarthritis, the constant cracking of knuckles will wear down the joints and the cushion supplied by the synovial fluid. "This means individuals who currently have osteoarthritis, caused by the breakdown of cartilage in the joints, could worsen their symptoms by cracking their knuckles repeatedly. However, cracking your knuckles plays no role in rheumatoid arthritis, which is caused when a person's own immune system attacks their joints. (UAMS Health)" There is no direct link to the development of arthritis when cracking your knuckles, but could worsen symptoms of those who have been diagnosed with it.
Arthritis can be developed through other factors. Arthritis is the inflammation of joints, it can cause pain, stiffness, reduced ability to move the joint, redness of the skin around the joint, etc. "There are more than 150 different types of arthritis. The cause of some forms is unknown, but arthritis can also be the result of disease, infection, genetic defect, injury, or overuse. Arthritis affects more than 50 million adults in the United States and is the nation's most common cause of disability. (Penn Medicine)"
Overall, there are many causes to the development of arthritis, but cracking your knuckles is not one of them.