I counted how many American presidents are depicted with facial hair of some form (sideburns, beard, mustache) and it turns out only 11 out of 44 were be-whiskered. There was a pretty good streak of 7 presidents starting with Lincoln (skipping A. Johnson and McKinley) to Taft with facial hair, but after Taft, it's all clean shaven presidents up to our current Mr. Obama.
The more accurate question would be, "Why did facial hair on American presidents suddenly become so popular in the mid-to-late-19th century?" I would feel comfortable saying that the reason was fashion, pure and simple. Take Lincoln for instance: he only grew his beard out when a young admirer mentioned it might make him look better. This tracks with a general trend in adult males towards facial hair that started around the time of Lincoln's presidency. It was only 9 years before little Grace sent her letter to the president that Schopenhauer wrote scathingly about the new trend of beards, calling them "a sexual symbol in the middle of the face, obscene..." and railing against them as an example of man reverting to his baser animal nature, as shaving was what set man apart from the animals. Schopenhauer's anti-beard stance didn't seem to have an impact, however, because by the 1870s facial hair was very much in fashion in America and Europe. Though there's been no study of this that I know of, one has only to look at images of adult men from 1870-1920 to see the range of facial hair present (the British and Russian royal families had some particularly noteworthy whiskers during this period).
So, to sum up, trends come and go, and presidents, like us all, are products of their time.
Edit: specifying American presidents.
hi! you may be interested in a couple of other responses
Why did facial hair fall out of favor among United States Presidents? - featuring /u/captmonkey
Why did facial hair fall out of fashion for American Politicians in Federal Politics? - featuring /u/chocolatepot