Let $x\in\partial M$ be any point on the boundary of $M$. Since $M$ is a manifold with boundary, there is an open neighborhood $U$ of $x$ that is homeomorphic to an open subset $V$ of $\mathbb H^n=\{ x\in\mathbb R^n : x_n\ge 0\}$ via a homeomorphism $\phi: U\to V$. Since $x\in\partial M$ we know that $\phi(x)\in V\cap\partial\mathbb H^n$, i.e. $(\phi(x))_n=0$. Since $\phi:U\to V$ is a homeomorphism, it restricts to a homeomorphism
$$\phi^{-1}(V\cap\partial\mathbb H^n) \to V\cap\partial\mathbb H^n.$$
We know that $V\cap\partial\mathbb H^n$ is open in the subspace topology on $\partial\mathbb H^n$ since $V$ is open in $\mathbb H^n$. Thus $V\cap\partial\mathbb H^n$ is an open neighborhood of $\phi(x)$ in $\partial\mathbb H^n$.

Now the open neighborhood $U\cap \partial M$ of $x$ in $\partial M$ is exactly $\phi^{-1}(V\cap\partial\mathbb H^n)$, which as we showed, is homeomorphic to an open subset of $\partial\mathbb H^n\cong\mathbb R^{n-1}$.

**On $\partial M$:** The most simple definition of $\partial M$ would be the set of all points of $M$ that don't have open neighborhoods which are isomorphic to open sets in $\mathbb R^n$. This gives you that any coordinate chart $\phi:U\to V$ of a manifold with a boundary has to send points of $\partial M$ to points of $\partial \mathbb H^n=\{x\in\mathbb H^n : x_n=0\}$, since all other points in $\mathbb H^n$ have open neighborhoods in $\mathbb R^n$ which can be pulled back to open neighborhoods in $M$ via $\phi$.

Another definition would be that $\partial M$ consists of all points of $M$ that are mapped to points of $\partial\mathbb H^n$ via all (or equivalently one) coordinate chart.