Suppose $E$ is an open set in $\mathbb{R}^n$. A differential form of order $k \ge 1$ in $E$ is a function $\omega$ symbolically represented by the sum $\omega = \sum a_{i_1 \cdots i_k}(x)dx_{i_1}\wedge\cdots\wedge dx_{i_k}$ where the indices $i_1,\cdots i_k$ ranges independently from $1$ to $n$which assigns to each $k$-surface $\Phi$ in $E$ a number $\omega (\Phi) = \int_{\Phi}\omega$ according to the rule
$$\int_{\Phi}\omega=\int_{D}\sum a_{i_{1},\dots,i_k}(\Phi(\mathbf u))J(\mathbf u)d\mathbf u$$ where $J$ is the Jacobian Matrix, $$J(\mathbf u) = \frac{\partial(x_{i_{1}},...,x_{i_{k}} )}{\partial (u_{1},....,u_{k})}$$
A $k$ surface in $E$ is a $\mathcal{C'}$-mapping $\Phi$ from a compact set $D \subset \mathbb{R}^k$ into $E$.
I really don't understand why are we defining a differential form in the following way and the fact that it assigns a number to each $k$ - surface does it have a intuition? Is that number unique?
I am very new to this field. An easy and elaborate answer would help me a lot . Thanks in advance!