It's hard to share intuition. But P(A|B) is a conditional probability. It is the probability of A, under the condition that B is given.
So in your example what has to be the condition that leads to such a high positive rate when using the test? The condition is that the patient actually has the disease. So it is indeed P(positive result | existing disease) = 80%
Now the probability of a positive result without any condition P(A) will probably be much lower. And the conditional probability of a positive result given a healthy patient P(A|C) will be even lower than that (hopefully if the test is good. More on that below).
Now imagine we have an actual result and want to figure out the probability that our patient is sick. Now the condition is the positive result. So it could be something like P(existing disease | positive result) = 20%
This would not be a good test because of a very high false positive rate, because apparently in a lot of cases we get a positive result without the patient actually having the disease.
Now imagine the following. These x all represent a person. The first row are sick people, the second row are healthy people. All of them get tested. Red indicates a positive result
xxxxx
xxxxxxxxxxxxxxxxxxxx
(20% of the population are sick)
Now 4 out of 5 people who are sick got a positive result -> P(positive result | existing disease) = 80%
But out of 20 people with a positive result, only 4 are actually sick -> P(existing disease | positive result) = 20%
Given the base rates above (20% of the population being sick), these two probabilites could actually be describing the same test. A test that gives a positive result in 80% of cases no matter what.
And it shows how important it is to not confuse the two. If you are interested in how to convert them, take a look at Bayes' theorem (but you probably know that).
I tried to describe it from a few different angles and say the same thing in different words, so that hopefully some of this helps you get a more intuitive grasp on the matter.
