The value of finger exercises for students of the piano, such as C-L. Hanon's The Virtuoso Pianist in Sixty Exercises, has been greatly debated. Many teachers claim the values of such exercises is limited or even lacking entirely. See, for instance, this interview with Graham Fitch: https://youtu.be/4bmY4FlNJJ4?feature=shared. In it, Mr Fitch, a renowned pianist and teacher, claims that piano finger exercises are pointless beyond the rudiments of playing or a convenient pattern for practising particular movements.
So, are finger exercises worth it? Do they help us, in a general way, to improve our piano playing and technique? In Hanon's case in particular, the claim is often made that it is monotonous and unnecessarily repetitive; that is, pace Fitch, we should focus on technical movements, and that, once these specialised finger exercises have been accomplished, further repetition is unnecessary.
Other classical pianists, like Annique Göttler, suggest that Hanon's finger exercises can form an essential part of regular piano practice, if used correctly. The difference here between Mr Fitch's specialised usage is her recommendation for regular use. Ms Göttler suggests that such finger exercises are useful not only for daily warm-ups, but also for establishing and maintaining a stable hand-position when playing the piano.
Hanon finger exercises strengthen hands and wrists
They may be called finger exercises, but in truth, you cannot strengthen your fingers, if only because there are no muscles there to strengthen. What you can do is develop good technical movements and posture at the piano so that you hands, wrists and arms support your fingers. You can also improve the muscles in you hand and wrist that support your fingers.
Some of these things, as Mr Fitch suggested, only require us to practise movements in isolation that we can then integrate into our playing of piano repertoire. Others may require a great deal of repetition, not only to build muscular strength but also to ensure that necessary movements can be applied unthinkingly at tempo. That is certainly not to say that Mr Fitch is wrong; many piano players may not require so much work to achieve the same, or better, results--every learner is different, after all.
Ms Göttler certainly claims they are useful to this end on her Youtube channel (https://www.youtube.com/@heartofthekeys). And I, for my part, have found them immensely useful over the years, both in the specialised usage previously mentioned and for daily practice.
Other benefits of playing Hanon finger exercises
There are many other potential benefits to regularly playing Hanon exercises. Indeed, if played through as Hanon himself recommended, they are excellent for developing and maintaining stamina. They also pre-accustom the tyro pianist to patterns and movements, such as trills, mordents and stretches that they will encounter in performance repertoire later on--especially, it has to be said, in baroque works.
However, in spite of these benefits, the experienced pianist may take or leave them as he or she chooses depending on their understanding of their own particular needs. The inexperienced piano player should listen to their teacher, who may recommend something else entirely.