Start with the definition of T*: (where the angled brackets denote the inner product).
Now ask yourself what does it mean to say that for all vectors y. If the first of those inner products is zero for all y it tells you that Tx must be zero. If the second inner product is zero for all y it tells you that x must be orthogonal (perpendicular) to the range of T*. You should be able to see from this that the kernel of T is {0} if and only if the range of T* is the whole space.