Question: 1 Here for "$i^*(c_j)$ form a basis for $H^*(F;R)$ in each fiber $F$", does it mean we see the $i^*(c_j)$s generate $H^*(F;R)$ as a $R$-module with each element in the form of $\sum_i\alpha_i$ with $\alpha_i\in H^i(F,R)$?
2 The last line claims that "weak homotopy equivalence, hence induces an isomorphism on all cohomology groups" but we only know $B$ is a CW-complex, we don't know if $p^{-1}(B^{n-1})$ is a CW-complex, thus we can't apply Whitehead theorem, so how can we deduce the the isomorphism on all cohomology groups?
All right, I will put it here as a proper answer.
Question 1. It means that for every fiber $F$ and its inclusion $i: F\to E$ generate $H^*(F;R)$ as an R-module - that is, every element may be written as $\sum_j a_ji^*(c_j)$.
Question 2. Weak homotopy equivalence induces isomorphisms on both homology and cohomology groups - look Hatcher's Algebraic Topology, Prop. 4.21.