Quite simply, I believe incest and supernatural appearances are enough to prove that "something strange is happening."  We are almost immediately told that Hamlet's uncle has married Hamlet's mother, not to mention the ghost that appears, telling Hamlet that his father was murdered by that very same uncle.  You've just got to love Hamlet's barb toward his mother meant to cut like a knife:  "A little more than kin, and less than kind!" (1.2.65).  It isn't long before Hamlet takes it further:

That it should come to this:  / But two months dead, nay, not so much, not two, / . . . a beast / Would have mourned longer--married with uncle, / My father's brother.  (1.2.137-138,150-152)

What better way to start a play off than incest?  If incest doesn't warrant the usage of the term "strange," then I don't know what would.  Hmmmm, except if a ghost appeared and told Hamlet that his uncle had murdered Hamlet's father in order to marry Hamlet's mother. 

Revenge his foul and most unnatural murder. / . . . But know, thou noble youth, / The serpent that did sting thy father's life / Now wears his crown. (1.5.25,37-39)

Thus stands Hamlet's pickle.  No wonder he "put an antic disposition on."  Ha!