So F_{*p} takes a vector X at p (derivation on the germs of smooth function at p) and sends it to a vector F_{*p}X at F(p) (derivation on the germs of smooth function at F(p)). So what is this vector F_{*p}X then? How does it act on a germ f at F(p)? This is what the formula is telling you: it says (F_{*p}X)(f) is just X(F*f). And this makes sense, since F*f =f o F is indeed a germ of functions at p.
