Why the government (and the DARPA in particular)is spending all these millions to improve the homomorphic encryption?which useful usage could it get?
In one sentence: data can be modified and calculated without being disclosed. Quoting from the Wikipedia page: ... effectively allowing the construction of programs which may be run on encryptions of their inputs to produce an encryption of their output. Since such a program never decrypts its input, it can be run by an untrusted party without revealing its inputs and internal state.
Like with most research agencies, their possible benefits aren't fully described. The full benefits of GPS, for example, were probably not imagined at the time either.
Homomorphic encryption is important for more than one reason.
However, the situations where I'm aware of its direct value is for self-modifying code e.g. packers and further protection of self-integrity-checking code. In other words, it's useful for software protection and potentially DRM.