<<many many babies start reaching out for food from the plate at the same time they start reaching out for toys!! >>

Babies investigative tools are their hands and their mouths.  Everything they reach for goes into their mouth, bits of food or paper or garbage on the floor are totally indistinguishable in appropriateness.  This is part of how they start learning the difference, among other things, between food and non-food.  "Some things mommy lets me put in my mouth.  Other things get taken away from me if I put them in my mouth.  Hmmm -- what's the difference."  

<<I have always wondered why a child reaching to take food from a parent
is a sign of interest in eating>>

It is also mimicking.  But it can become an interest in eating  when we do let them put foods into their mouths.  They start developing awareness of tastes and textures of food.  If we let them put any food they want to into their mouths, they may end up being much more adventuresome eaters than others who are restricted to "baby" foods only.  Or not.

Chanita, San Francisco