Why exactly is this a problem?
Great question. I don't have a great answer. Not everyone sees the current situation as a problem, which is copyright is the way it is today.
Here is what I think, and from that, others perhaps will understand why I think the current situation is unreasonable.
Intellectual property, like property, is a complete social fiction - its a very useful one, but nonetheless - a fiction.
Property is a big unspoken social agreement we have that assigns resources to individuals and entities and gives them superior rights of control over those resources. This assignment we call "ownership", and is a critical part to nonviolent resource distribution with many independent entities. In civil society it is simply given that this property mapping of things to people/organizations is "real", but in fact it is only supported, like all rules, (both laws and social mores) if people generally agree - both agree that the rules are reasonable, and agree that they each will (in the vast majority) follow those rules. If people don't agree, laws don't work.
Intellectual property extends the idea of this big shared social mapping of resources (property) to intangible "intellectual" creations (written words, music, video and most anything translatable into computer bits). The basic idea of intellectual property says that if one entity (person, company) did a lot of work in creating something, they should have superior rights to control it for a while. By itself, this is a very reasonable idea.
On the other hand, there is no physical basis to support property rights on information objects like there are on working land or creating physical things. Many would argue extremely convincingly that in a highly connected world, most people would be much better off if there were no intellectual property at all. That only those large organizations profiting from culture creation and limiting access to culture would be those harmed by eliminating IP entirely.
However, most important to the debate from my perspective is one of culture. The shared actions of humans that create the beauty, education, entertainment, and everyday existence for human beings is now encoded very often in digital information used to create experiences we all share. The fundamental question at hand is this: are we better off with human experience owned by corporations, or not? To me, this is the essence of the whole copyright debate - it has nothing to do with the specifics of law or legality, the politics of lobbying groups, or even the money people make off IP - it has to do with what kind of entity gets to create and control human culture, and whether it happens primarily by and for individuals in an open way, or whether it happens primarily under corporate ownership in a closed way.
Currently, we unequivocally have the latter. Large corporations primarily own the most valuable and most widely shared cultural elements in all 1st-world countries. The length of copyright basically only benefits and perpetuates corporations now. Governments with WIPO and other treaties are trying to enforce long, strong copyright protections globally. Its not individuals' creative expression driving how we live, how we think, how we get news and information, how we are entertained, how we are educated - but rather (and I'm being extremely general here) - it is corporations. These statements are extremely broad and there are many counter examples, but I'm referring to the largest factors and the most momentum in society.
I see it as unreasonable that culture created today will never be available to me openly and legally in my lifetime. The only reason the system works this way is because large companies profit more from IP working this way than other ways. The social fiction of IP is no longer a good deal for the individual in this case. This basic understanding that this legal fiction is no longer a good deal for individuals is why so many people redistribute music, why black market activity on IP is out of control, and why some large cultures simply reject entirely the premise of strong IP protection.
As I said above, not everyone sees this as a problem. I have no particular negative bent against companies, they are essential to human progress. I do have a problem when the interests of the corporation squashes the interests of individuals. In the case of shared human culture, who owns it, and who controls it - corporate interests seem to trump individual interests most every time.