Imagine the universe has a centre that regularly produces new stars which then drift away at a constant speed. This has been going on forever, so there are infinitely many stars. We can label them by age, or equivalently by their distance from the centre: star 1 is the youngest, then comes star 2, then star 3, and so on, without end. The stars in turn produce planets at regular intervals. So the older a star, the more planets surround it. Today, something happened to one (and only one) of the planets. Let's say it exploded. Given all this, what is your credence that the unfortunate planet belonged to the first 100 stars? What about the second 100? It would be odd to think that the event is more likely to have happened at one of the first 100 stars than at one of the next 100, since the latter have far more planets. Similarly if we compare the first 1000 stars with the next 1000, or the first million with the next million, and so on. But there is no countably additive (real-valued) probability measure that satisfies this constraint.
Another example (inspired by an example in Kit Fine's "Counterfactuals without possible worlds"). Infinitely many rocks are positioned on a slope: rock 1, rock 2, rock 3, and so on. Rock 1 is firmly kept in place by concrete. Rock 2 is slightly more loose, rock 3 slightly more loose still, and so on. Exactly one of the rocks has just rolled down the slope. Again, it would be odd to think that the falling rock is more likely to be among the first 100 than among the second 100, or among the first million than among the second million, given that the earlier ones are more firmly held in place. But this is what countable additivity requires.