Is there an economic interpretation of the production transformation function?
It is, as you say, a mathematical device to define the production set in a way that is analytically tractable

I have seen one description as thinking of it as a measure of the amount of technical progress now required to make the combination $y$ a feasible one, so 


if $F(y)=0$ then that combination is just feasible with current technology, 
if $F(y)\lt 0$ that combination is feasible with less than current technology, 
if $F(y)\gt 0$ that combination requires more than current technology enables and so is currently not feasible


though this looks more like handwaving and rationalisation rather than something actually meaningful