This claim is true. Not only from personal experience but from reading many articles it is proven that movements and hobbies make a happy life. An article written by Utah State University by Emma Parkhurst, Professional Practice Extension Assistant Professor states says that having hobbies will reduce stress, enhance well-being, improve social connection, decrease depression and anxiety, as well as overall improve mental health.
Behind the claims that hobbies improve mental health, there was also proven reasoning behind each claim. "One study showed approximately 75% of participants’ cortisol levels (one of the most widely studied hormonal markers of stress) were lowered after making art." the article said. As well as, "hobbies that include physical activity have been linked to many health benefits including reduced stress, reduced blood pressure, and a lower heart rate."
This article also addresses the issue of not having time for hobbies. And being able to make some. Overall this claim is very true, and something you can even prove yourself. Overall, the data provided gives strong context to why this claim is true.
https://extension.usu.edu/mentalhealth/articles/how-hobbies-improve-mental-health