This claim is an exaggeration of a possible correlation, and wrongfully implies causation. At first glance, the article by UCLA Health describes that the overuse and repeated motions we use our hands for in phone activities is associated with thumb pain and inflammation - both of which can be early symptoms of arthritis. However, although the author spends a lot of time exploring the link between texting and arthritis, they state that "texting is a relatively new activity", and arthritis "takes decades to develop", so the relationship between the two "remains to be seen". Ultimately, UCLA Health displays to the audience that while there may be a correlation between excessive hand motions with hands and developing arthritis, correlation does not mean causation.
Another article that I found to factcheck this claim was by Yale Medicine, titled "Is There A Link Between Hand Pain and Smartphone Use?" This article investigated the pains associated with excessive smartphone use, and what different people reported to feel. It stated that it is "not yet clear if excessive phone usage can be the cause" of conditions like arthritis, it may be a source of aggravation to pre-existing pains. Overall, it seems that excessive phone use can lead to irritation of existing pains, but there is no evidence to prove a direct link.