The problem is that people want an answer to the question, "Here is a pile of data, what should I believe?" But mathematically, the proper role of data is to modify existing beliefs, and not to dictate beliefs.
Statisticians can spend forever explaining this. But instead have gone with the cop-out of asking a question that is confusingly similar to the one that people want to ask. It is popular exactly because it is so easily misunderstood. You can give any number of lectures on what it actually means - I guarantee that it will be misunderstood.
Worse yet, p-values are sensitive to particulars of experimental design that logically should never matter to inferences. My aunt and uncle provide a classic example. They wanted a son and a daughter. They had 6 sons then a daughter. Are they biased towards one gender? The p-value for this question is .03125 (we'd see this much evidence against with all boys, all girls, last girl, last boy, so 4/2^7). But if I instead said that they planned to have 7 children, and had 6 sons then a daughter, the p-value changes to 0.125 because the one child out could have been anywhere in succession. But in Bayes' formula, their plans if something else had happened cannot ever affect how we adjust our inferences.
(This example is not made up. When Lorna found that they had a daughter, she told Bill to get a vasectomy from the delivery table. By another crazy coincidence, their 7 children were also born on all the days of the week.)