In a standard cheap-talk setting, a sender (S) has better information on a state of the world and wants to communicate this information to a receiver (R) who then takes an action. However, S and R prefer different actions conditional on the state.
Importantly, S is free to send any message independent of what she knows. That is, she cannot commit to a signal given the true state. This is a complicated strategic setting, and communication may fully break down - in a babbling equilibrium any message is ignored, and R decides based on his prior.
Even equilibria with information transmission can be bad for S.
KG give S more commitment power in the sense that she can design a signal structure and the signal realization is truthfully communicated to R. This commitment assumption helps to see this as a sort of mechanism design problem. Instead of setting rules on transfer payments, the designer manipulates the informational environment. In this problem of information design, the commitment to the signal structure determines the meaning of a signal such that "babbling equilibria" are circumvented.
Can R be "persuaded" to do something S wants when R is a rational Bayesian and well aware that S wants to manipulate him? Perhaps surprisingly, the scope for manipulation is quite large. KG show how to optimally do this. The prosecutor-judge example illustrates the paper very nicely.
Essentially, KG interpret the signal structure as a distribution over posteriors and a signal realization then determines R's posterior. Can S just use any such distribution? No, S is restricted to a Bayes plausible distribution, the expected posterior must be equal to the prior! I guess that is why the "Bayesian" is important. Of course, R being a Bayesian is not a unique feature and quite standard, but in the same sense this word shows up all the time when analyzing standard settings, think of BNE, PBE, BCE...
However, I don't like the term persuasion so much. When I think of persuasion, misrepresenting and concealing information is part of it. This is explicitly excluded in KG! Depending on the application, I find the assumption on S's commitment power quite a stretch for a persuasion. If you just consider the theory without a persuasion application in mind, a smooth way around a discussion about commitment power is to consider a setting with symmetric information: S doesn't know the state herself but designs the information structure.