Let $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary.$^1$
We know that there is a countable family $((\Omega_i,\phi_i))_{i\in I}$ of $k$-dimensional $C^1$-charts$^2$ of $M$ with $$M\subseteq\bigcup_{i\in I}\Omega_i.$$
We know that there is a $k$-dimensional boundary $C^1$-atlas$^1$ $((\Omega_i,\phi_i))_{i\in I}$ of $M$ for some $I\subseteq\mathbb N$.
Let $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. Note that $(\mathbb H^k)^\circ=\mathbb R^{k-1}\times(0,\infty)$ and $\partial\mathbb H^k=\mathbb R^{k-1}\times\{0\}$.
Let $B$ denote the closed unit ball in $\mathbb R^k$, $B_+:=B\cap(\mathbb H^k)^\circ$ and $B_0:=B\cap\partial\mathbb H^k$.
Why can we choose $((\Omega_i,\phi_i))_{i\in I}$ such that the manifold interior$^3$ $\Omega_i^\circ$ is equal to $\phi_i^{-1}(B_+)$ and the manifold boundary $\partial\Omega_i$ is equal to $\phi_i^{-1}(B_0)$?
$^1$ i.e. each point of $M$ is locally $C^1$-diffeomorphic to $\mathbb H^k$.
If $E_i$ is a $\mathbb R$-Banach space and $B_i\subseteq E_i$, then $f:B_1\to E_2$ is called $C^1$-differentiable if $f=\left.\tilde f\right|_{B_1}$ for some $E_1$-open neighborhood $\Omega_1$ of $B_1$ and some $\tilde f\in C^1(\Omega_1,E_2)$ and $g:B_1\to B_2$ is called $C^1$-diffeomorphism if $g$ is a homeomorphism from $B_1$ onto $B_2$ and $g$ and $g^{-1}$ are $C^1$-differentiable.
$^2$ A $k$-dimensional $C^1$-chart of $M$ is a $C^1$-diffeomorphism from an open subset of $M$ onto an open subset of $\mathbb H^k$.
$^3$ $x\in M^\circ$ if and only if there is a $k$-dimensional $C^1$-chart $(\Omega,\phi)$ of $M$ such that $x\in\Omega$ and $\phi(\Omega)$ is $\mathbb R^k$-open.
$x\in\partial M$ if and only if there is a $k$-dimensional $C^1$-chart $(\Omega,\phi)$ of $M$ such that $x\in\Omega$ and $\phi(x)\in\partial\mathbb H^k$.