A handwaving approach.
Suppose we represent a decaying state by a wave function
In order for this state to ``decay away'', the density 
must
decrease as a function of time. This in turn requires that the energy
is complex: 
.
Then:
and hence
Thus the exponential decay law is a consequence of a complex energy.
Now, the original state 
is not an eigenstate of the system.
It turns out that it is spread out in energy.
Expand it in terms of eigenstates of the energy E:
Fourier transforming this:
where 
.
The state thus has a Lorentzian distribution in energy.
Thus the energy of a decaying state is not an eigenvalue of the system nor a constant: in particular, the energy of the state is distributed over a region with a width determined by the decay constant.