What is the significance of $A+ (B\cap C)=(A + B)\cap C$, where $A\subseteq C$, for modules? My book (Introduction to Ring Theory, Paul Cohn) states this as a theorem and gives a proof. The book usually skips over trivial/easy proofs, so I don't really understand why this is in here.
Isn't the statement absolutely obvious for sets $A,B,C$ with $A\subseteq C$? You just need to draw a diagram and it becomes immediately obvious! 
Is there a reason why it would not be obvious in modules?
 A: I'm not sure if the statement is obvious to you but in general $+$ and $\cap$ don't distribute over each other. I wouldn't expect them to either since $\cap$ is a set theoretic operation and $+$ is only defined for modules/ideals/abelian groups. However, when $A\subset C$ they do distribute over one another.
But the proof shouldn't go something like this:
$$A+(B\cap C)=(A+B)\cap (A+C)=(A+B)\cap C $$ where the last equality comes from $A\subset C$ because here we assumed they distribute in the first place which is false in general. So any proof of this fact can't use the above method (or the other direction).

An explicit example that these operations don't distribute is by taking the submodules of the $\mathbb{Z}[X]$-module $\mathbb{Z}[X]$ where we can now just look at ideals
$A=(2)$, $B=(x-1)$, and $C=(x+1)$
then $$A+(B\cap C)=(2)+(x^2-1)=(2,x^2-1)$$
and $$A+ B \cap A+ C=(2,x-1)\cap(2,x+1)=(2,x+1)$$
since $(2,x-1)=(2,x+1)$.
A: Judging from your diagram, it looks like (?) you are interpreting $+$ as set union. Certainly, if you are working with sets using union and intersection, your picture is accurate. 
But using using the $+$ to join submodules of a module does not behave the same way. If $A$ and $B$ are submodules of a module, $A+B$ contains many more elements than just $A\cup B$ in general.
To summarize the situation, the property proved is that the lattice of submodules of a module (join $+$ and meet $\cap$) is a modular lattice. The lattice of subsets of a set, on the other hand (join $\cup$, meet $\cap$), is also modular, but even more strongly it is a distributive lattice. In general, the lattice of submodules of a module does not have to be distributive.
For more about modularity, let me refer you to a question we had some time ago on the topic: Why are modular lattices important?
