The production set has a simple meaning: It is the set of all production vectors that are feasible to a firm.
The production function also has a simple meaning: It gives the output quantity for a given vector of input quantities (for a firm that produces only one output. )
But what is the production transformation function? It is defined as the function $F(y)$ such that the production set is $\{y: F(y)\leq 0 \}$. Essentially, it is a way to encode the production set.
But is there also a meaningful interpretation of this function itself, rather than merely as a mathematical device to define the production set in a way that's analytically tractable?