What exactly are confidence intervals?

So, a friend (who would probably prefer to remain nameless) is currently looking for jobs in finance, and sent me some sample interview questions that they like to ask people. One of them is the following:

A drug trial gives the result that the drug works better than the
placebo, with 95% confidence. What exactly does this statement mean? What further assumptions are needed to be able to deduce that the probability of the drug working is actually 95%?

Now, I wasn't entirely sure I knew the answer to this question. I thought I did, but despite having studied rather a lot of statistics, I don't think anyone ever actually told me what a 95% confidence interval was, so I did what everyone does faced with such a situation, and checked the wikipedia article. Wikipedia is singularly confusing on the matter, but it gives the answer as roughly the following.
"The drug works better than the placebo with 95% confidence" means "if there were no difference, and I did the test lots of times I would expect to get these results less than 1 time in 20" or, more simply:

P(results|no difference) < 0.05

What we actually want is to make a statement about how likely there is to be a difference. In fact, according to the question, we want to be able to say:

P(no difference|results) < 0.05.

I have no idea how having the answer to one question is supposed to help us answer the other (we could apply Bayes theorem, but it's unlikely to give us the same value for both probabilities).

What is more surprising, though is that pretty much no-one acknowledges that there's a difference! People regularly say things along the lines of "there's a 95% chance that the drug is better than a placebo" (eg, this page which starts by saying that '"Significance level" is a misleading term that many researchers do not fully understand', and then goes on to say that a statement which is true with 95% significance 'has a 95% chance of being tru' - perfectly illustrating the confusion).

I've never really understood the difference between bayesian statistics and frequentist statistics before. Bayesians use something sensible called credible intervals, which *do* allow them to make statements like the one in the previous paragraph. I can't see why anyone would prefer the frequentist version, but I don't really know enough about it to have a properly informed opinion. Anyone?