"Enjoy!" Enjoy what?

Enjoy is used as an absolute verb here, which means that the object is implied but not stated.

There is another famous example also: if looks could kill... Here, the transitive kill is used as an absolute verb.

Though, there is a history behind the good wish "Enjoy!" meaning "Enjoy your meal!" and it is derived from a Yiddish phrase. An article in The New York Times talks about the history of this phrase and mentions the first absolute use of the transitive enjoy:

In his 1986 book, "Yiddish and English," Random House lexicographer Sol Steinmetz cites this 1968 quotation of furrier Jacques Kaplan by Marylin Bender in The New York Times: "It's a dancing over the volcano attitude, an enjoy-enjoy philosophy." That reduplication is typical of Yiddish -- Es, es means "Eat, eat" -- and the friendly command of Enjoy! comes from Hob anoe , a Yiddish phrase derived from the German hob , "have," and the Hebrew hanoe , "enjoyment."

The earliest example of the absolute use of the transitive enjoy comes from an essay by English author John Ruskin in "The Eagle's Nest," in 1872: "It is appointed for all men to enjoy, but for few to achieve."

http://www.nytimes.com/1992/05/24/magazine/on-language-enjoy.html