Where did the word “investigate” originate and What does investigate mean?

Hunters through all the ages have known that their game could be found by following its footprints or the traces that it left of its passage.

Roman hunters used the word vestigium for footprint or trace, whence our word vestige, trace, and thus investigo meant to trace or track out, as a dog scents game.

Our word investigate is not quite so literal in meaning, but the term does imply the thoroughness of search similar to that of a dog on a cold trail.