My nutshell version is this: suppose we start with a small world of perfectly happy people, then add a totally separate additional set of people not quite as happy, but with lives still worth living. It seems to the author of the paradox, Derek Parfit, that it would be implausible for this to be worse overall, there being a greater sum of total happiness. But the end of many such additions of addition sets of people, and their also their unifications (which he also hold harmless), is the repugnant conclusion of unending masses of people with lives barely worth living.
Parfit interprets this paradox as a disproof of any kind of utilitarianism, stemming from the notion that the "better than" relationship is not transitive. (See link above.)
A variant of utilitarianism has a different way of explaining this paradox, much preferable to my mind. This is called "average utilitarianism." It claims that the utility cannot be summed over individuals, only averaged. Thus the addition of people of lower happiness LOWERS the critical average level of happiness, so the addition of less happy people to the world makes it less well off overall, even if their lives are worth living.
I much prefer the "average utilitarian" resolution of this paradox.
In fact, I think one can go beyond the "average utilitarian" resolution. Addition of any number of people of less than the highest possible happiness may be detrimental, even if their level of happiness is greater than the existing average. Perhaps one way to see this is to imagine the reverse of happiness, call it "sadness", and say the point of a moral philosophy should be to reduce the total amount of sadness.
One problem with this kind of disutilitarian analysis is that it proposes as the perfect world one with the least number of people that can be perfectly happy. There could be another world with even more perfectly happy people.
I guess I really don't have a problem with that for several reasons. One is that human existence is by its very nature an encroachment upon other forms of existence, such as animal wildlife. Although it is problematic to figure how to include these other forms of existence in the sum of sadness, it ought to be recognized that currently imaginable human existence has a destructive footprint on other living species, and therefore is best kept as small as possible. From that perspective, the smallest human world of perfectly happy people might already be too large.
And one can quibble over "perfect happiness". What I mean, actually, is here is not absolute perfection, but happiness not limited by lack of resources or lack of possible human relationships. The smallest such "perfect" world could not be one person because people are not like that; they need relationships, and even such things as division-of-labor to achieve the greatest possible happiness.
OK, rather than try to quibble so much, perhaps I can merely suggest that both the "average utilitarian" and the "average disutilitarian" analyses are biased, but the best we could do would be to aim for some level of population between them.