Why does someone stop eating?

There are many reasons that someone might stop eating. In long term care (aka nursing homes), the interdisciplinary team (all the people involved from all disciplines - nutrition, social services, nursing, infection control, activities, the doctor and anyone else on the team who can add insight) , the Resident (aka patient) and their family needs to be involved. Are they depressed? In pain? Is there a urinary tract infection (or any infection) that is affecting their appetite? Is medication making everything taste "bad" or "funny"? Are they weak? There are so many reasons someone stops eating. Sometimes their disease processes are affecting them in ways no one has realized. With input from everyone (and not everyone will agree on what they see), the doctor can order testing and physically assess the patient.
A sore mouth, ill fitting dentures - the causes are many, and not always easily found. But, with observation, persistence, and the input of many different perspectives, the cause can often be found.
Sadly, sometimes the body systems are failing, and the body is responding by decreasing appetite. It isn't something that can always be fixed - but, sometimes (more often than not) a cause can be determined, and interventions will be effective.
People don't stop eating just because they are getting old. Never accept that as a reason.