Not sure if the answer will be clear, but that's inherent in the weird and counter-intuitive nature of the concept itself (which I consider as a major disadvant

Not sure if the answer will be clear, but that's inherent in the weird and counter-intuitive nature of the concept itself (which I consider as a major disadvantage of it). It is an interval that was obtained using a confidence interval generation procedure, for which the following is true: for any true value of the quantity you're estimating, the procedure yields an interval that contains that correct value 95% of the times, when run on different random samples collected conditioned on that true value. You want to estimate some quantity, let's say the cancer risk ratio of hair-dyers to non-hair dyers. You make some (noisy) experimental measurements on people. Then, let's say someone hands you a confidence interval generating procedure. You feed it with your measurement data and it spits out an interval. To claim that this interval is a 95% confidence interval is actually a claim about the confidence interval generating procedure, not about the particular interval. And that claim is that for any true cancer risk ratio value, if we ran lots of experiments and fed all the results to this procedure, then 95% of the resulting intervals would contain the correct corresponding value. So in more detail: Assume the true ratio is, say, 1.05. Then if we repeated our noisy experiment in a 1.05-ratio world and applied the confidence interval generating procedure to the results, we would expect to get 95% of those intervals containing the number 1.05 and 5% of them not containing it. If we assume the ratio is 1.2, the same thing should hold. For this analysis to work, we need an idealized measurement model that tells us what the distribution of measurements looks like given any particular underlying value of the quantity of interest (say the cancer ratio). Then using this model and the description of a given confidence interval generating procedure, we may be able to prove mathematically that it indeed has the above property. Most of the time, statisticians use standards procedures, for which this property has been proved and the proof is widely known. They rarely invent new such procedures. So, in summary: we don't know what the true value is in reality. But we can prove that the confidence interval generating procedure has the above explained property. Then, we will call the interval that this procedure gives us on our real measurement data a 95% confidence interval. What it definitely does not mean is that the true value is in the interval with 95% probability, and that's the most common misunderstanding. I'm not sure if this helps at all. These things are really counter-intuitive and twisted against our natural way of thinking.