Understanding integration along fibers I understand the definition of integration of a differential form along fibers as it is stated in Wikipedia article as follows:

Let $\pi :E \rightarrow B$ be a fiber bundle over a manifold with compact oriented fibers. If $\alpha$  is a $k$-form on $E$, then for tangent vectors $w_i$'s at $b$, let
$ (\pi _{*}\alpha )_{b}(w_{1},... ,w_{k-m})=\int _{\pi ^{-1}(b)}\beta  $
where $\beta$ is the induced top-form on the fiber $\pi^{-1}(b)$; i.e., an $m$-form given by: with $\widetilde{w_i}$ lifts of $w_i$ to $E$,
$$
\beta(v_1, \dots, v_m) = \alpha(v_1, \dots, v_m, \widetilde{w_1}, \dots, \widetilde{w_{k-m}}).
$$

However, I don't intuitively understand why it is defined like this and don't know why it is called integration along fibers ?
 A: In the comments, OP mentions integrating the restriction to a fiber of a differential form. If $\omega$ is an $n$ form, and the fiber $F$ of $\pi:X\to B$ is of dimension $k$, then $\int_F \omega$ will be $0$ unless $n=k$. When $n=k$, OPs expectation is right; $\int_{X/B}\omega$ is the function on $B$ given by
$$
b\mapsto \int_{\pi^{-1}(b)}\omega.
$$
This formula isn't going to give us anything when $n\neq k$. When $n>k$, there is something we can do: I prefere to think of the integration along the fiber as a special case of the currential pushforward. For a full development of the theory of currents, I think deRham's book is still the best source. All I'll say is this: We can define a continuous functional $\pi_*(\omega)$ on the vectorspace of forms on $B$ (suitably topologized) by
$$
\pi_*(\omega)(\tau) = \int_X \omega\wedge\pi^*\tau.
$$
This works for any proper and oriented smooth map $\pi$, but must be reinterpreted if $X$ isn't itself oriented, so let's just assume that $X$ and $B$ are oriented too. The deal with integration along the fiber is that in the special case that $\pi$ is a fiber bundle, there exist a smooth $n-k$ form $\int_{X/B}\omega$ on $B$ such that
$$
\pi_*(\omega)(\tau)=\int_B\left(\int_{X/B}\omega\right)\wedge \tau.
$$
The existence of the form $\int_{X/B}$ is a generalisation of the Fubini theorem: Suppose $\pi$ is trivial, i.e. isomorphic to the projection $F\times B\to B$. Then the Fubini theorem says something like:
$$
\pi_*(\omega)(\tau)=\int_{F\times B}\omega\wedge\pi^*\tau = \int_B \int_F \omega\wedge\pi^*\tau
$$
Since $\pi^*\tau$ is constant with respect to the $F$ coordinates, we can move it outside $\int_F$, and we get
$$
\pi_*(\omega)(\tau)=\int_B\left( \int_F\omega\right) \wedge\tau.
$$
So $\int_F\omega$ is the form I was denoting $\int_{X/ B}\omega$ before. Using a partition of unity subordinate to a cover of $B$ which trivialise $\pi$, I think we could prove that $\int_{X/B}\omega$ exists in general.
To conclude: You should think of fiberbundles as being kind of like products, but globally more twisted. Fubini's theorem is for actual products. But of course there must be something to say for fiberbundles with twisting to! That's integration along the fiber.
