Understanding of pullback $f^*\xi$ in manifolds I will translate the passage of the article available here for a good understanding of provided concept I want to grasp:
Due to vector space structure in a fiber $\xi_p$, $p \in M$, the functions $s$, $\tilde{s}$ belonging to section $\Gamma(\xi)$, $\varphi \in \mathcal{C}^{\infty}(M)$ always allows $s+\tilde{s}$ i.e. $\varphi(s)$ with domain $G_s \cap G_\tilde{s}$. In case function $f$ maps from $\mathcal{C}^{\infty}$-manifold F into $\mathcal{C}^{\infty}$-manifold M, hence we have a vector bundle $f^*\xi$ induced by function $f$ from vector bundle $\xi$.
I cannot picture an image like tangent vector bundle $T_p F$ or $T_{f(p)} M$. May you?
I understand its name recall some reverse operation, such that a vector bundle $\xi$ on manifold M drawbacks a vector bundle $f^*\xi$ on base manifold $F$. Do I understand it correctly?
 A: You mentioned vector space structure so I'll work in the context of vector bundles, so given a map $f: F \rightarrow M$ and a bundle $\xi$ with projection $\rho: \xi \rightarrow M$, what is a visualisation of the pullback bundle $f^*\xi$?
Well, roughly speaking, a vector bundle on a space, such a manifold, is given by attaching a vector space to each point of the manifold (there is a certain local trivialization condition that must hold but let's not worry too much about that for now).
Here is an sketch of this (via the first of two questionable paint drawings), given a point $x \in M$ let $V_x$ denote the vector space attached at the point $x$, i.e. $V_x = \rho^{-1}(x)$:

Now, we want to make use of this family of attached vector spaces to points of $M$ and attach a vector space to each point of $F$, we do this is as follows.
Given a point $y \in F$, we apply $f$ to get the point $f(y) \in M$. Since $f(y)$ lies on $M$ there is a vector space attached to it, namely $V_{f(y)} = \rho^{-1}\left(f(y)\right)$. Now all we do is declare that we are attaching $V_{f(y)}$ to $y \in F$, the picture looks like this:

What we have done here is found an attachment of vector spaces to points of $F$, this is what is known as $f^*\xi$. By the way, from the second picture you can see why it is called the pullback bundle, you are in some sense pulling the vector space back along $f$ so you can attach it to a point of $F$.
If $\pi: f^*\xi \rightarrow F$ is the projection of this new-found bundle on $F$, then we have $\pi^{-1}(y) = V_{f(y)}$ i.e the vector space attached to $y \in F$ is the same vector space attached to $f(y) \in M$.
Finally, you should read (a lot more formally of course) about this and how the local trivialization condition holds for this construction, but this is the general idea.
