I have the following lemma and proof in my lecture notes ($\mathcal{R},\mathcal{N}$ denote range, kernel, respectively, and $\mathcal{X}$ and $\mathcal{Y}$ are Hilbert spaces).
I was hoping there was a simpler proof: for any linear operator between Hilbert spaces $B:X\to Y$, we have $X=\mathcal{N}(B)\oplus \overline{\mathcal{R}(B^*)}$, so I believe we have, for $A,B\in \mathcal{L}(X,Y)$, $$\overline{\mathcal{R}(A^*)}\subseteq \overline{\mathcal{R}(B^*)} \iff \mathcal{N}(A)\supseteq \mathcal{N}(B) $$ Thus, to prove $\overline{\mathcal{R}(A^*A)}\supseteq \overline{\mathcal{R}(A^*)}$, we only have to prove $$\mathcal{N}(A^*A) \subseteq \mathcal{N}(A)$$ which is easy, because if $A^*Ax=0$, then \begin{align*} \langle A^*Ax,x \rangle &=0\iff \langle Ax, Ax \rangle = 0, \end{align*} so $x\in \mathcal{N}(A)$.
Does this work?