Why do the sequent calculus NOT left and NOT right rules work?
You could start by considering simplified versions of the rules and build intuition by considering those cases. For instance,

$$\frac{\neg A, \ B \implies C}{B \implies C, \ A} (\neg L)$$ 

can be interpreted as stating that $(\neg A\wedge B)\Rightarrow C$ implies 
$B\Rightarrow (C\vee A)$. 

So if it is the case that $\neg A$ and $B$ are true implies $C$ is true, then if only $B$ is true, either $C$ is true (independently of $\neg A$) or $A$ is true, for otherwise $\neg A$ would be true, which combined with $B$ being true would make $C$ true.

A similar game can be played with the other case.