Why do poles in the left half of the S plane make a system stable?
An $n$th order linear system is asymptotically stable only if all of the components in the homogeneous response from a finite set of initial conditions decay to zero as time increases, or mathematically written:
$\displaystyle{\lim_{t \to \infty}}\sum_{i=1}^{n}C_ie^{p_it}=0$ ($p_i$ are the system poles).  As time increases (in a stable system) all components of the homogeneous response must decay to zero.If any pole has a positive real part there is a component in the output that increases without bound, causing the system to be unstable. So, in order for a linear system to be stable, all of its poles must have negative real parts (they must all lie within the left-half of the s-plane).  An "unstable" pole, lying in the right half of the s-plane, generates a component in the system homogeneous response that increases without bound from any finite initial conditions.