Thank you.Sigh :-(. Is this again from Griffiths's QM textbook? I cannot understand how an expert of the subject can be so sloppy and confusing for students. That's the more ununderstandable since he's obviously a brillant teacher of physics, as one can get from reading is electromagnetics textbook and many articles in AJP.
To understand "decay", particularly the exponential-decay law (which is necessarily an approximation only), you need to consider time-dependent perturbation theory. The standard treatment is known as Wigner-Weisskopf approximation. Here's a very clear derivation on the example of spontaneous emission, but it's of course applicable very generally