Attaching two manifolds along their boundary

EDIT 6/17/19: Another reader recently pointed out that the argument I sketched below is not appropriate for this place in the book, since it uses properties of compactness that are not introduced until Chapter 4. I'll leave my original answer here for those who are curious; but a much better solution is to just abandon the idea of arranging for the images of the coordinate maps to be product open sets, because that was not really needed anyway. I've added a correction to my online list.

Original Answer:

Here's a way to see it. Choose charts $(U,\varphi)$ and $(V,\psi)$ such that $x\in U$ and $y\in V$, and let $U_1 = U\cap \partial M$ (which is a neighborhood of $x$ in $\partial M$) and $V_1 = V\cap \partial N$ (a neighborhood of $y$ in $\partial N$). After shrinking $U$ if necessary, we may assume that $h(U_1)\subset V_1$. (Because $U_1\cap h^{-1}(V_1)$ is a neighborhood of $x$ in the subspace topology $\partial M$ inherits from $M$, there is an open subset $\widetilde U\subset M$ such that $U_1\cap h^{-1}(V_1) = \widetilde U\cap\partial M$. Replacing $U$ by $U\cap \widetilde U$ does the trick.) Let $U_0\subset U_1$ be a neighborhood of $x$ that's precompact in $U_1$, and let $V_0 = h(U_0)\subset V_1$, a precompact neighborhood of $y$ in $V_1$.

Now let $\widehat U=\varphi(U)$, $\widehat V=\psi(V)$, considered as subsets of $\mathbb H^n = \mathbb R^{n-1}\times [0,\infty)$. Since $\varphi(\overline {U_0})$ is a compact subset of $\mathbb R^{n-1}\times \{0\}$, there is some $\varepsilon_1>0$ such that $\varphi(\overline {U_0})\times[0,\varepsilon_1)\subset \widehat U$. Similarly, there's $\varepsilon_2>0$ such that $\psi(\overline {V_0})\times[0,\varepsilon_2)\subset \widehat V$. Now just take $\varepsilon = \min(\varepsilon_1,\varepsilon_2)$, and replace $U$ by $\varphi^{-1}\big(\varphi(U_0)\times[0,\varepsilon)\big)$ and $V$ by $\psi^{-1}\big(\psi(V_0)\times[0,\varepsilon)\big)$.