The abstract of the paper is far more informative that then article. Functional encryption allows to decrypt a ciphertext c with a function (there can be many)

The abstract of the paper is far more informative that then article. Functional encryption allows to decrypt a ciphertext c with a function (there can be many) f and get F(c), without revealing anything else about c. This was previously doable where F was public in a way that could be plausibly efficient. This was an interesting result because it might lead to things like efficient searchable encryption without using very slow fully homomorphic encryption(FHE). A lot of these applications, however required F to be obfuscated. This paper achieves hiding f, for a somewhat weak notion of obfuscation, and more crucially, by using fully homomorphic encryption(FHE). Given that FHE is effectively (very really but very inefficient) cryptographic pixie dust, it's not too surprising you can do this. However, for a lot of the applications for functional encryption, you could do int with FHE in other ways.