Suppose throughout that $M^n$ is a manifold with two conformal Riemannian metrics $\widetilde{g} = e^{2\phi}g$.
The volume form is a tensor, so it suffices to work pointwise. At a point $p$, in normal coordinates $(x^1, \ldots, x^n)$ for $g$ at $p$ one has $(g_p)_{ij} = \delta_{ij}$, whence $(\widetilde{g})_{ij} = e^{2\phi}\delta_{ij}$. Thus at $p$, \begin{align*} \mathrm{vol}(\widetilde{g}) &= \sqrt{\mathrm{det} \widetilde{g}}\, \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n \\ &= \sqrt{(e^{2\phi})^n} \,\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^n \\ &= e^{\phi n} \mathrm{vol}(g). \end{align*}