This was predominately for lung aliments, specifically tuberculous which was not well understood in the 19th and early 20th century. These locations were popular because of the elevation found in the Alps and the perceived health benefits of the 'clean' air in those locations. Alpine Italy and Switzerland were popular, but many at the time also travelled to similar alpine regions of France and Germany. I apologize for the rather succinct response as I'm at work, but please take a look into my sources if you'd like to know more.
Warren, Peter. "The Evolution of the Sanatorium: The First Half-Century, 1854-1904." Canadian Bulletin of Medical History 23.2 (2006): 457-76.
David L. Ellison, Healing Tuberculosis in the Woods (Westport, CT: Greenwood Press, 1994), p. 75; and Paul Ehler, Hatte ich nur meine Bekannten schon begrusst… Gesichte der Sanatoriums-Frischuft-Liegekuhr 1860-1950 (Switzerland: Nebelspater-Verlag Rorschach, 1982), p. 10.