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First, let me say beyond knowing that group cohomology comes from derived functors and all the properties that come with it, I don't have much intuition for general group cohomology. However, in number theory there are a number of places in which you can realize the cohomology groups as parametrizing some other objects you are interested in. To understand these other objects (which you could feasibly have very real intuition about) it then suffices to use the machinery of group cohomology. Below are a few examples. I forget to put continuous subscripts on everything, so beware. Mistakes are mine.

First, in the realm of (edit:local) class field theory we have the Brauer group $Br(K)$ which is the abelian group (under the tensor operation) of central simple algebras over $K$ up to the equivalence $A \sim M_n(A)$. It turns out that there is a natural isomorphism between $Br(K)$ and $H^2(G_K, \overline{K}^{\times})$. This isomorphism provides two worlds in which to make important calculations. For instance, the statement that every central simple algebra over $K$ splits over an unramified extension of $K$ may be proved explicitly using the Brauer group or it may be proven by checking that $H^2(G(K^{nr}/K),\overline{K}^\times) = H^2(G_K,\overline{K}^\times)$. From either proof you are able to obtain a proof of the other. Perhaps, also in the realm of class field theory, the local Artin map arises as a map on Tate groups given by a certain cup product but that would take a bit longer to explain. You should look at Milne's notes on CFT for all of this (Ch III and Ch IV for what I have said).

Here is another. Suppose that $B$ is a topological ring and $G$ is a topological group and $G$ acts continuously on $B$. Then, we consider finite free $B$-modules $X$ equipped with a semi-linear action, i.e. $g(bx) = g(b)g(x)$ for all $b \in B$ and $x \in X$. It turns out that all such objects are parametrized by $H^1(G,GL_d(B))$. (Warning: this is non-abelian cohomology, so this is only a pointed set. The point corresponds to the trivial semi-linear representation $B^d$ with the diagonal action.) This comes up very early in the part of $p$-adic Galois representations where one studies the period rings $B_{dR}, B_{HT}$, etc.

Finally, consider the situation where one has a representation $\overline{\rho}:G_{\mathbb Q,S} \rightarrow GL_n(\mathbb F_p)$ and one wants to know whether it has litings $\rho: G_{\mathbb Q,S} \rightarrow GL_n(R)$ where $R$ is some complete DVR with residue field $\mathbb F_p$. If we already have a lifting to $GL_n(R')$ and $R$ and $R'$ are nice enough (there is a surjection $R \rightarrow R'$ whose kernel $I$ is killed by the maximal ideal in $R$) then the obstruction to lifting further lies in the cohomology group $H^2(G_{\mathbb Q,S},I\otimes Ad(\overline{\rho}))$ where $Ad(\overline{\rho})$ is the vectorspace $M_n(\mathbb F_p)$ together with conjugate action by $\overline{\rho}$. This (I've specialized a few things) is written down in Mazur's paper "Deforming Galois Representations".

To finish, I will just say what my adviser told me when I asked him a similar question as you are asking: "Just wait and you will see how much clearer your thought can become with group cohomology."

1

First, let me say beyond knowing that group cohomology comes from derived functors and all the properties that come with it, I don't have much intuition for general group cohomology. However, in number theory there are a number of places in which you can realize the cohomology groups as parametrizing some other objects you are interested in. To understand these other objects (which you could feasibly have very real intuition about) it then suffices to use the machinery of group cohomology. Below are a few examples. I forget to put continuous subscripts on everything, so beware. Mistakes are mine.

First, in the realm of class field theory we have the Brauer group $Br(K)$ which is the abelian group (under the tensor operation) of central simple algebras over $K$ up to the equivalence $A \sim M_n(A)$. It turns out that there is a natural isomorphism between $Br(K)$ and $H^2(G_K, \overline{K}^{\times})$. This isomorphism provides two worlds in which to make important calculations. For instance, the statement that every central simple algebra over $K$ splits over an unramified extension of $K$ may be proved explicitly using the Brauer group or it may be proven by checking that $H^2(G(K^{nr}/K),\overline{K}^\times) = H^2(G_K,\overline{K}^\times)$. From either proof you are able to obtain a proof of the other. Perhaps, also in the realm of class field theory, the local Artin map arises as a map on Tate groups given by a certain cup product but that would take a bit longer to explain. You should look at Milne's notes on CFT for all of this (Ch III and Ch IV for what I have said).

Here is another. Suppose that $B$ is a topological ring and $G$ is a topological group and $G$ acts continuously on $B$. Then, we consider finite free $B$-modules $X$ equipped with a semi-linear action, i.e. $g(bx) = g(b)g(x)$ for all $b \in B$ and $x \in X$. It turns out that all such objects are parametrized by $H^1(G,GL_d(B))$. (Warning: this is non-abelian cohomology, so this is only a pointed set. The point corresponds to the trivial semi-linear representation $B^d$ with the diagonal action.) This comes up very early in the part of $p$-adic Galois representations where one studies the period rings $B_{dR}, B_{HT}$, etc.

Finally, consider the situation where one has a representation $\overline{\rho}:G_{\mathbb Q,S} \rightarrow GL_n(\mathbb F_p)$ and one wants to know whether it has litings $\rho: G_{\mathbb Q,S} \rightarrow GL_n(R)$ where $R$ is some complete DVR with residue field $\mathbb F_p$. If we already have a lifting to $GL_n(R')$ and $R$ and $R'$ are nice enough (there is a surjection $R \rightarrow R'$ whose kernel $I$ is killed by the maximal ideal in $R$) then the obstruction to lifting further lies in the cohomology group $H^2(G_{\mathbb Q,S},I\otimes Ad(\overline{\rho}))$ where $Ad(\overline{\rho})$ is the vectorspace $M_n(\mathbb F_p)$ together with conjugate action by $\overline{\rho}$. This (I've specialized a few things) is written down in Mazur's paper "Deforming Galois Representations".

To finish, I will just say what my adviser told me when I asked him a similar question as you are asking: "Just wait and you will see how much clearer your thought can become with group cohomology."