In the $10$-th chapter of the text Introduction to Smooth Manifolds written by John M. Lee it is written what put to follow.
The boundary of a smooth manifold with corners, however, is in general not a smooth manifold with corners (think of the boundary of a cube, for example). In fact, even the boundary of $\overline{\Bbb R^n_+}$ itself is not a smooth manifold with corners. It is, however, a union of finitely many such: $\partial\overline{\Bbb R^n_+}=H_1\cup\dots\cup H_n$, where
$$
H_i:=\{(x^1,\dots,x^n)\in\overline{\Bbb R^n_+}:x^i=0\}
$$
is an $(n-1)$-dimensional smooth manifold with corners contained in the subsapce defined by $x^i=0$.The usual flora and fauna of smooth manifolds -smooth maps, partitions of unity, tangent vectors, covectors, tensors, differential forms, orientations and integrals of differential forms- can be defined on smooth manifolds with corners in exactly the same way as we have done for smooth manifolds and smooth manifolds with boundary, using smooth charts with corners in place of smooth charts or generalized charts. The details are left to the reader.
In addition, for Stokes's thorem we will need to integrate a differential form over the boundary of a smooth manifold with corners. Since the boundary is not itself a smooth manifold with corners, this requires a special definition. Let $M$ be an oriented smooth $n$-manifold with corners, and suppose $\omega$ is an $(n-1)$-form on $\partial M$ that is compactly supported in the domain of a single oriented smooth chart with corners $(U,\varphi)$. We define the integral of $\omega$ over $\partial M$ by
$$
\int_{\partial M}\omega:=\sum_{i=1}^n\int_{H_i}(\varphi^{-1})^*\omega
$$
where each $H_i$ is given the induced orientation as part of the boundary of the set where $x^i\ge 0$. […] Finally, if $\omega$ is an arbitrary compactly supported $(n-1)$-form on $M$, we define the integral of $\omega$ over $\partial M$ by piecing together with a partition of unity just as in the case of a manifold with boundary.
So unfortunately I did not completely understand the definition of the integral of $\omega$ on $\partial M$: in particular if the coordinate patch $\varphi^{-1}$ is defined in an open set $U$ of $\overline{\Bbb R^n_+}$ then why the integral
$$
\int_{H_i}(\varphi^{-1})^*\omega
$$
is well defined if generally $U$ is not contained in $H_i$? Anyway it seems to me that Lee is implicitly claiming that the boundary of a $n$-manifold with corners is equal to union of $n$ $(n-1)$-manifolds with corners so that he integrates $\omega$ over each such manifolds but I am not sure about. After all $H_i$ is a $(n-1)$-manifolds with corners contained in $\overline{\Bbb R^n_+}$ and $U$ is there open so that $U\cap H_i$ is an open set of $H_i$ and thus in particular it is a $(n-1)$-manifold with corners so that the integral
$$
\int_{U\cap H_i}(\varphi^{-1})^*\omega
$$
is well defined but I am not sure that it is equal to the first. Anyway $(\varphi^{-1})^*\omega$ is compactly supported and its support is contained in $U$ that is open in $\Bbb R^n$ or in $[0,+\infty)^n$ so that it can be exteded to a smooth function defined in $\Bbb R^n$ and so in $H_i$ and thus effectively the integral
$$
\int_{H_i}(\varphi^{-1})^*\omega
$$
is well defined. So is the last a good explanation? Moreover I am not sure that $(\varphi^{-1})^*\omega$ can be extended to a smooth $(n-1)$-form defined in $\Bbb R^n$. So could someone help me, please?
I point out I did not really study Lee's book but I only consulted it for pratical reasons -i.e. I am studying Continuum Mechanics and so I am ran into integration on manifolds with corners. However I studied the integration on manifolds (with boundary) on the text Analysis on Manifolds by James Munkres but it seems that Lee formalism can be easily adapted to that of Munkres and vice versa. So I put here some reference about Munkres formalism for an easily comunication.
Best Answer
First, on an $n$-manifold with corners there may be more or less than $n$ boundary hypersurfaces. Consider e.g. $M=[0,1]^2$. It is true however, that $M=\mathbb{R}_+^n$ has exactly $n$ boundary hypersurfaces.
What is going on in the integration is just that first, $\omega$ is pulled back via the chart to yield an $n-1$-form $\alpha = (\varphi^{-1})^*\omega $ on some open set $U\subset \mathbb{R}_+^n$. You are right in saying that $U$ is not contained in any hypersurface $H_i$. But that is no problem, you can just restrict $\alpha$ to $U\cap H_i$ (aka: pull back via the inclusion map $U\cap H_i\rightarrow U$) to get an $n-1$-form on $U\cap H_i$, for which integration is well defined.