It'd be Popper's work on Falsifiability that they're referencing. Not sure I agree with the text you quote as it seems to get the wrong end of the stick with the principle and the swans.
In general Popper stated that for something to be "scientific" it had to be falsifiable, i.e. there is a means and way to prove it wrong. You may never prove it wrong, but there is a way to do so and that would be the feature of experimentation.
So if a theory is proposed, it should also be identified how it can be falsified. For example, "global warming" whether it be the general principle of the earth warming or that it is caused by man-made activities can be falsified. We can either show a) there is no warming or b) there is, but due to natural causes. Therefore, global warming theory is scientific because there is a means to falsify the theory.
Not to be controversial, but in general "God exists" cannot be scientific because you cannot falsify that statement. There is no means of proving that something doesn't exist, you can't prove a negative.
Without getting too much into the philosophical aspects, the swan analogy demonstrates a couple of aspects to science.
First, you observe a white swan and you state "that swan is white". We know that of all swans in existence, at least one swan is white. However, Popper's view is that this is pretty much all we can state. It comes back to a classic mistake made with some issues where a correlation is deemed to be a cause. So we observe an effect (say cancer in a population) and also identify that all the people who have cancer have at some point consumed white bread. That's a correlation, we cannot conclude that whiet bread causes cancer.
However, making the statement from seeing one white swan that all swans are white is weak science, but it is falsifiable. Simply observing one black swan at some point would disprove the statement. So to confuse, it's a weak conjecture, but it is in theory still falsifiable, i.e. not great but we can work with it and study it. However, the means of falsifying it would involve observing every single swan on the planet which is so impracticable as to make it unfalsifiable.
Therefore Popper's main argument was that he really didn't like the above "conjecture" means of developing scientific theory. He wanted more practical experiment based means of flasifying. The "Swan Philosophy" is really exploited in the world of conspiracy theories as while on ther surface there is a means of falsifying, it isn't practical.
So someone may say all swans are white and I find a black swan on the canal. They then change their statement to "all swans are white except those on the canal". Using that kind of conjecture, you can always modify your original theory to remain valid despite of evidence, largely because it is impracticable or impossible to fully falsify the original theory.
In summary the swan philosophy demonstrates both the good observation and theorising aspects of science, but also the potential for weak conjecture. It demonstrates the limits of falsification and that in order to meet a criteria as a scientific theory, those proposing the theory must also demonstrate or at least identify how their theory could be falsified.