I don't think there's a dot product there (except implicitly, in the definition of the gradient.) Looking at the equation applied to your case, we'd have
As I see it, the way to interpret the LHS of this equation is to take the vector , rotate by left-multiplying by , then you stuff the result of the rotation into the function , and finally, you take the gradient.
I interpret the RHS of this equation as follows: first, you take the gradient of , and then you evaluate that function at the point , and finally you left-multiply by
If I were you, I'd work out a simple example in, say, 2 dimensions. See how it works out.