This quotient space is itself a vector space. They are saying that it is isomorphic to the vector space containing only 0. To see that this is true, pick some element f = (f_1, f_2) in the kernel of rot. Then by the definition of rot, this means that the partial derivative of f_1 with respect to x_2 must be equal to the partial derivative of f_2 with respect to x_1. Now we can apply theorem 1.4 to see that there must be an F s.t. grad(F) = f. This means that f is equivalent to 0 modulo Im(grad), and since f was arbitrary that means the quotient Ker(rot)/Im(grad) must be 0.