I had the same problem. In fact I do not understand the statement
Confusion 1: What does it mean by "global cohomology classes $e_1,\ldots, e_r$ on $E$ when restricted generate the the cohomology of the fiber"
A cohomology class in $H^*(E)$ lies in some $H^n(E)$ and and hence is a closed $n$-form. Why are we allowed not to specify the superscript, $*$?
Confusion 2: In the linked proof by jdc why can we say that $H^*(F)=\Bbb R \{e_k \}$ what does this even mean?
1) Suppose that there is a sequence of cohomology classes $e_i \in H^*(E)$, with $1 \leq i \leq r$; we may take the restriction of these classes to the fiber, $i^* e_i \in H^*(F)$. The demand is that the set $\{i^* e_1, \cdots, i^* e_r\}$ is a vector space basis for the real vector space $H^*(F)$.
It is true that $H^*(F)$ is a graded vector space, but that's not the point here. Each of the classes $e_i$ may as well be chosen each to lie in some specific $H^{k_i}(E)$, and then the claim is that the if $i_1, \cdots, i_r$ is the set of indices for which $k_i = k$, the vector space $H^k(F)$ has a basis given by $\{i^* e_{i_1}, \cdots, i^* e_{i_r}\}$. This is just slightly more cumbersome to state.
2) As far as I can tell he is writing $\Bbb R\{e_k\}$ as shorthand for the set $\Bbb R\{i^*e_1, \cdots, i^*e_r\}$, writing $e_k$ for a generic element of that set of generators.