I'm fairly new to topology, and so far I've understood cohomology via cochains. 

First we build an object called a cochain ($C^n$), then define a differential map that takes you from $C^n$ to $C^{n+1}$. Then all the types of cohomology groups I've encountered can be easily defined as:
\begin{equation}
H^n(M) = Z^n / B^n
\end{equation}
Where $Z^n$ is the n-cocyle and $B^n$ is the n-coboundary.

The trouble I'm facing is in defining the map $\partial^n: C^n \rightarrow C^{n+1}$

In other cohomologies, the map has an intuitive explanation. For example, in deRham cohomology, the map just sends a differential form to a higher differential form, but due to Stokes' theorem we can interpret the nth form being the boundary of an n+1th form. In simplical homology, we define the map to be the boundary of the n-chain, so a line segment will have the two endpoints as its boundary etc.

In group cohomology however, the map is defined as: $f:G^n \rightarrow A$ where A is a G module, and:
\begin{equation}
\partial^i f(g\_0,\cdots,g\_i) = g\_o f(g\_1,\cdots,g\_i) + \sum\_{j = 1}^i (-1)^j f(g\_0,\cdots, g\_{j-1} g\_{j}, g\_{j+1}, \cdots, g\_i) + (-1)^{i+1} f(g\_0,\cdots, g\_{i-1})
\end{equation} 

Now I understand that you can prove the various "standard" properties of this map using this definition (like $\partial^{i+1} \partial^i f = 0 $ etc.), but what is the intuitive explanation for what's going on? Why is this map defined the way it is? How does this correspond to the boundary interpretation of other types of cohomologies? How can I "see" it being a boundary without explicitly computing $\partial^2 = 0$? 

And what happens when the groups are continuous? For lie groups can we apply deRham cohomology? What about cohomology on principal G-bundles? Any introductory resources on this subject would be much appreciated!