Assuming that you are dealing with real matrices, note that if $A^T A mathbf{x} = mathbf{0}$, then $A mathbf{x}$ must be in the null space of $A^T$, which in particular is orthogonal to the row space of $A^T$ (and hence orthogonal to the column space of $A$). But $A mathbf{x}$ is by construction in the column space of $A$!

All this is to show that $A mathbf{x}$ is both in the column space of $A$ and the space orthogonal to the column space of $A$. These two facts together imply that $A mathbf{x} = mathbf{0}$.