Threats made over Twitter (a.k.a the Internet) are by nature federal crimes if the person making the threat is from another state or country. Relevant statute is 18 U.S. Code § 875 - Interstate communications.
( c ) Whoever transmits in interstate or foreign commerce any communication containing any threat to kidnap any person or any threat to injure the person of another, shall be fined under this title or imprisoned not more than five years, or both.
It is thus not surprising at all that the FBI rather than the SFPD would handle the case.
The "people" who "dug around" a little are /v/ and /pol/ of 4chan, and the corresponding crowd at Reddit. The characterizing trait of those people is that they dont worry whatsoever about being either prudent or diligent or fair. They just want to see sh*t go down, because wow that's entertaining.
Case closed.
ETA: In fact, on closer reading, a threat from the same state over Twitter is also a federal crime, since it's made "in insterstate [...] commerce". Any threat over Twitter is a federal crime.
