I've seen this around a lot. I can't find much that's very official or very clear, so I might be missing something; but I think this can be said as first approximation:
(1) The rejection of limbo (to the extent it is one) is not, in fact, a very strong rejection. All that appears to have happened is that the Pope has approved of a commission's argument that children dying before the age of reason are granted the beatific vision.
(2) This, if it is what was really done, is not a rejection of limbo. It is a common misconception that limbo is for unbaptized children. In fact, limbo is just sheol, and I can pretty much guarantee you that the Pope has not claimed that there is no sheol, or anything equivalent to it.
In fact, limbo, generally, is a very easy doctrine to prove from Scripture. Sheol is mentioned over and over again in the Old Testament; it is translated as hades in the New Testament; and it is found in the doctrine of Christ's descent into hell -- i.e., into hades. What is difficult to say is who is in limbo, and why, and what's involved in being in limbo. Traditionally, limbo was considered a state reserved for two groups in particular -- the unbaptized righteous -- patriarchs like Samuel (1 Sam 28) and Abraham (Luke 16) who had prefigured Christ's coming; and the unbaptized innocent, like children. The usual understanding of it was that it was a state of natural happiness, without torment or penalty. Further, people can be redeemed from limbo, whereas they can't from hell in the strictest sense; Christ's descent into hell, in fact, is in part the claim that Christ redeemed the patriarchs from limbo and exalted them to a supernatural happiness.
Two big and controversial questions have been whether noble pagans are admitted into the state of limbo and whether unbaptized children who were not of God's chosen people could be redeemed from it. Dante suggested that there was a limbo of noble pagans; although, of course, this may have been as much a poetic as a theological move. The point that has really worried people, however, is whether unbaptized children in limbo can be redeemed from it. Part of this has just been confusion; people don't distinguish properly between hell as limbo and hell as second death, in part because of our tendency to polarize the afterlife rather artificially into Heaven (all Beatific Vision) and Hell (all torment and regret), without considering that hell (and its cognates) used to have a much broader meaning than they usually do today, covering all states of immortality that do not involve the supernatural grace of the Beatific Vision. The two are not even remotely the same; limbo is just what you get if you can't ever be happier than we can be in this life, and hell as second death is punishment in a positive sense. On the other hand, we have very little information about how far the redemption we know the patriarchs received extended. Much of the historical controversy over the limbo of children was due to the Calvinist tendency to point at it as yet another instance where the Catholics had shown human weakness and Pelagianized grace. (If I recall correctly, the Pelagians, in fact, were among the ones who early on used the term 'limbo'.) It is a sign of the times that the pendulum of controversy has swung entirely in the opposite direction, and limbo is treated as yet another case of Catholics delighting in the punishment of the innocent.
Of course, because people worried so much about the limbo of children, when people talk about the doctrine of limbo, they usually mean the doctrine that all unbaptized children after death are in a state of limbo, which is not quite the same thing. For, after all, the possibility of limbo as a general state is Catholic dogma; the Council of Florence, for instance, makes very clear that souls of those who depart in original sin receive the proportionate punishment (which includes at minimum lack of Beatific Vision); and the Council of Trent is very clear that the unbaptized are not free of original sin. What is not Catholic dogma is the view that unbaptized infants (or anyone else) are never redeemed from the state of limbo by any extraordinary work of Christ and God's grace; and the usual claim has always been that we simply lack information about such matters. So limbo in that sense -- in the sense of unbaptized infants being eternally unredeemed -- is what is really at issue here.
Such is the first approximation, anyway. I find it noteworthy that all the news announcements are in future tense up to October 5, when the Pope was 'expected' to do something about it -- the Pope had not taken action on it -- and then, nothing. It's also noteworthy that they've been saying he was just about to do it for well over a year now. So the real state of affairs seems to be this: there's no information yet, and we can't really say anything until the information actually comes up. And yet people keep saying things....