The verb enthuse, formed by backformation from the adjective enthusiastic, means to be enthusiastic or to show enthusiasm. The word peeves some people who care about English usage, but it has been widespread in the language long enough to gain at least grudging acceptance. No one is forced to use it, but it is certainly a word, and most English speakers know what it means.
Similar backformations such as donate (from donation) and diagnose (from diagnosis) are now unquestioned. Whether enthuse will gain a similar level of acceptance is impossible to say, but the word has been around for well over a century, so it could be that the issue is already settled.
With enthuse, however, problems arise where enthused, as a participial adjective,takes the place of the perfectly good enthusiastic. Of course, there’s nothing wrong with having two words that mean the same thing, but enthused tends to carry an ironic tone, at least in the U.S., and is difficult to use in earnest.
Just for fun, below is an ngram graphing occurrence of enthuse and the clunky phrase be enthusiastic about in the large number of English-language texts in Google’s books index. The graph spans 1850 to 2000.
In these examples, enthused appears where enthusiastic would work just as well:
Not only am I sold on the new Pentastar engine, I am totally enthused on all the new Chrysler products. [letter to Globe and Mail]
I was never too enthused about big bookstores like Borders. [comment on Washington Post]
So when word circulated that Lee would become consulting chef for Eden South Beach, Miami locals were naturally enthused. [Miami New Times]
And in these, enthuse is easier to justify because it’s used as a verb (although those who dislike enthuse would surely argue that there are other words that would work better in these instances):
Way to the other side of the political spectrum, Pat Buchanan has enthused about the “Tea Party Tory” now in Number 10 Downing Street. [The Algemeiner]
It was particularly amusing to watch all the models enthuse about how CSI is one of the longest-lived franchises on TV. [AV Club]
Charities will be able to bid for contracts worth millions, they enthuse. [The Economist]