I am beginning to learn cohomology, namely in the small and functional appendix of Silverman's book, The Arithmetic of Elliptic Curves. While the theory is efficiently presented and I have no problem understanding the formal arguments, I am facing serious intuitive issues.
The first group of cohomology of a $G$-module $M$ is defined building on the notions of cochains (functions from $G$ to $M$), the cocycles (these functions satisfying a cochain condition) and coboundaries (some subset of those of specific form). These names suggest a strong geometric intuition behind them however I do not catch anything intuitive in introducing these notions. The main problem seems for me to be that if we have the exact sequence $$0 \to P \to M \to N \to 0,$$
we only have the following partial sequence for the $0$-th cohomology groups: $$0 \to P^G \to M^G \to N^G$$
and we would like to complete it to a long exact sequence. Why do we introduce the notions above to do so? (without saying: a posteriori it works, so it should be worth it) And what intuition should I have behind the cohomology groups?