Thanks very much for your interest (assuming the existence of some "you")!

"It is a truly wonderful fact...that all animals and all plants throughout all time and space should be related to each other in group subordinate to group." (From On the Origin of Species, p.128)
Darwin didn't think that the Linnaean system was junk; he thought it was "a truly wonderful fact." This is Darwin saying, "I love Linnaeus!" The reason that we still use Linnaeus's system today is because Darwin himself endorsed it.
Clearly, something else has gone awry with the received view. As we've already seen, Linnaeus didn't follow from Aristotle; now we see that Darwin didn't make this clean break from Linnaeus. The true story is that Linnaeus made a clean break with Aristotle and then Darwin followed from Linnaeus.
In fact, one way to read "On the Origin of Species" is as giving an explanation for why it should be true that all the world's organisms are organized within these nesting groups, as biologists call them. Evolution gives rise to the Linnaean system through what Darwin called the "principle of divergence."
This is, quite famously, the only figure to appear in the "Origin"--it's in chapter four, in case you're interested--and it's intended to illustrate Darwin's principle. To briefly explain what we're looking at: each branch here represents a population, the dotted lines represent lines of descent, and the horizontal lines represent some number of generations. As species evolve, some populations will develop traits and branch off--diverge--from other populations, eventually creating new species that will then have diverging populations, and so on.
Eventually, the pattern of divergence will give us groups within groups, just as the Linnaean system says. So we may consider each blue circle a species, each red circle a genus, and the green circle an order, for instance.
But how do we know when one of these groups represents a species and when it represents a genus, or an order, and so on? According to Darwin, the thing that separates the divisions in Linnaeus's system from one another is the degree of divergence: that is, the further up on this tree you go, the more divisions will arise.
So we start out at the bottom with species; as we move up, genera appear; further on up, orders come out. Incidentally, if this looks tedious, it's because it was. I wish that I got paid by the circle.
But there's a problem. Darwin says that "all plants and all animals throughout all time and space" should belong to each of Linnaeus's categories. Looking at the tree above, organisms in the groups at the bottom belong to species, but not to genera or orders and so on, because there aren't any genera or orders at that point. Our chart should look like this:
In this version, every group belongs to each and every one of the categories, but this also can't be the way things are according to Darwin's principle. Organisms in the groups at the bottom can't belong to species and to genera, for example, because the difference between those categories is in their degrees of divergence and there hasn't been any divergence yet. But Darwin also says that each organism has to belong to both groups. Very simply put, Darwin has contradicted himself.
So, what does this mean for the two problems in philosophy of biology that I mentioned at the outset? Well, as it turns out, those problems are entirely bound up with the way you resolve this contradiction. There are really only two ways out: you can either reject the Linnaean system or you can say that the scale of Darwin's tree is relative.
Modern biologists tend to reject Darwin's principle of divergence and consequently reject Linnaeus's system as truly representing the way the world should be carved up. But this presents a serious problem: after all, it should be pretty obvious that there are natural divisions. There is far more in common between you, me, and Natalie Portman than there is between any of us and a chimp, for example. At least, I would hope that if Ms. Portman had a choice between a chimp and me she wouldn't have to think too hard about it. To be clear, I would hope that her obvious choice would be me and not the chimp. But if Linnaeus's system isn't giving us the real divisions within nature, then how do we draw that boundary between humans and chimps? That's the species problem.
On the other hand, maybe you want to preserve Darwin's principle of divergence. If that's the case, though, then the only way to explain how organisms in the groups on his tree both must and can't be in all of the Linnaean categories is to say that the groups don't necessarily represent groups of organisms. In fact, this is precisely what Darwin says: he says that the branches on the tree might represent species or genera; he says that the horizontal lines might represent thousands of generations or hundreds of thousands or, hell, let's even make it millions of generations. But if each group can be a species or a genus or an order or whatnot, it's also worth noting that these groups are definitely unified and bounded off from one another--they're individuals. That means that natural selection can work on them. That's the levels-of-selection problem.
What I intend to have shown here is that these problems have come out of the historical development of biological theory; consequently, we need to know the truth of that history in order to develop adequate solutions to the problems. The true story certainly isn't the one that's been commonly received: we didn't go from Aristotle to Linnaeus, then have all of that scrapped because Darwin. Instead, Darwin's work is a consequence of the Linnaean tradition, which itself was a clean break from Aristotle's work.
The best way to resolve Darwinian problems is to understand from where Darwin was coming. With an improved understanding of that history, my hope is that we can formulate better solutions. I also hope that this discussion might inspire some of you to look into the matter a little more closely and perhaps find some of those solutions.
Thank you.