Lewis and Short's definition of interimo
suggests that in this word "inter" has its usual meaning of "among" or "between": to take out of the midst, to take away, do away with, abolish; to destroy, slay, kill.
In their entry for inter
, Lewis and Short list "intereo" and "interficio" as examples of "inter" meaning "under, down, to the bottom". But in their entries for these two verbs they seem to abandon that idea.
their definition has "inter-" meaning "among": to go among several things, so as no longer to be perceived.
From that definition it's an easy leap to "to be lost", "to perish", etc.
And their entry for interficio
suggests a logical transition from "to put between" to "to devour" (to put between jaws?), "to destroy", and "to slay".