This is true. The simple explanation is this: For calculating the decay rate of an excited state, you use Fermi's Golden Rule, which involves the matrix element
$$|\langle f | V | i \rangle|^2$$
where $f$ and $i$ denote the final and initial state, respectively.
Since the final state contains the electron in its groundstate together with a photon created by this decay, the nature of your cavity determines what the matrix element will be:
For example, if your cavity forbids standing waves of the emission frequency, decay is suppressed. The study of these effects goes under the name of Cavity QED
I would not say that this is due to the Casimir effect. Rather, this effect and the Casimir effect are both due to the boundary conditions created by your plates.
I don't exactly know who first studied this. I suggest consulting a review article such as this one.