The crucial concept which is necessary for the integration of differential forms on manifolds is the parametrization of the manifold. Actually, this also needed for "normal integration" over a set $M \in \mathbf R^k$, furthermore vectors tangent to the parameterized manifold are needed. Let's go step by step.
- Math on manifolds need charts and in general one chart only covers part of the manifold, we actually need a covering of the manifold $\{ \cal U_i\}_{i\in \mathbf N}$, then we can write for a n-form $\omega$:
$$\omega = \sum_{i\in \mathbf N} h_i \omega$$ with $$\sum_i h_i =1$$
where $h_{i\in \mathbf N}$ is a partition of 1. The partition is chosen in a way that for each $i$ a chart can be chosen that covers completely $\cal U_i$.
In the following we will always assume that only one chart is needed by arguing that if more charts are needed instead of $\omega$ $h_i\omega$ is considered.
This was just the prelude.
- Now we assume a parametrization $g: \mathbf{R}^n\rightarrow \cal U_i \in \mathbf{R}^k$ respectively $\in M\,$ (n<k). At each point of the grid on $\cal U_i$ smooth tangent vectors $(\xi_{1,i_1},\ldots, \xi_{n,i_n})$ can be easily obtained by ($\Delta x_{i_1}$ and $\Delta x_{i_n}$ are the $x_{1}$- respectively $x_{n}$-intervals of the grid in $\mathbf{R}^n$ where $i_1, \ldots, i_n$ the indices which enumerate the grid points)
$$ \xi_{1,{i_1}} = (\frac{\partial g_{x_1}}{\partial x_1}, \ldots, \frac{\partial g_{x_k}}{\partial x_1})|_{g(x_1,\ldots ,x_{n})}\,\,\Delta x_{i_1}$$
$$ \ldots $$
$$ \xi_{n,{i_n}} = (\frac{\partial g_{x_1}}{\partial x_n}, \ldots, \frac{\partial g_{x_k}}{\partial x_n})|_{g(x_1,\ldots ,x_{n})}\,\,\Delta x_{i_n}$$
The differential form $\omega$ res. $h_i\omega$ is evaluated on these tangent vectors in the following form:
The n-form is evaluated on the tangent vectors $\xi_{1,i_1},\ldots,\xi_{n,i_n}$ at all grid points $(g(x_{i_1},\ldots, x_{i_n}))$ of $\cal U_i$ and for the evaluation of the integral
$$\int_{\cal U_i} h_i\omega \approx \sum_{i_1} \ldots \sum_{i_n}h_i\omega(\xi_{1,i_1},\ldots \xi_{n,i_n})$$
Riemann sums have to built up from the resulting values $\sum_{i_1} \ldots \sum_{i_n}\omega(\xi_{1,i_1},\ldots \xi_{n,i_n})$. If everything is done correctly, one ends up with an approximate integral where even the Jacobi determinants (which are needed for integration in k-dimensional space) are included automatically (n-forms are made like to produce the Jacobi determinant automatically!) and the only missing step is to take the limit $ \Delta x_{i_1} ,\ldots , \Delta {x_{i_n}} \rightarrow 0$ to obtain the value of the integral. A nice property of this definition is that the integral is independent of the parametrization $g$ which is chosen. So actually the form can be evaluated on any tangent vectors $\xi$ as long they can be associated to a smooth parametrization $g$ (or $f$ etc.).
In this explanation I tried to be rather general, it is more intuitive to imagine for instance a 2-form over a surface $M$, in that case $n=2$ and $k=3$. This is a very common application in electrodynamics.
The point 2) actually can be formulated more elegantly by the concept of a pullback, where integral of $\omega$ over $M$ is written like:
$$\int_M \omega := \int_D g^{\ast} \omega$$
where D represents the area in $\mathbb{R}^n$ which is mapped by the parametrization $g: D \rightarrow M$ (or one of the partitions $\cal U_i$) and $g^{\ast}\omega$ is the pullback of $\omega$. This implies also mapping (the differential of $g$) of the tangential vectors of $D$ to $\xi_{m,{i_m}}$ with $m=1,\ldots, n$. (in other words the tangent vectors $\xi_{m,{i_m}}$ are the push forwarded tangent vectors of $D$) $D$ is a (flat)subset of $\mathbb{R}^n$, so its tangential vectors lie inside $\mathbb{R}^n$ which solves one of your doubts. The definition of integrals of differential forms on manifolds $M$ via the pullback of the form on a subset $D$ of $\mathbb{R}^n$ is the mathematically correct definition. It also should be familiar to you.
Last, but not least: An important prerequisite for this to work is that the manifold $M$ is orientable. In physics this is (practically) always the case.
