The suits were based on the questionable notion that making files available in directories through peer-to-peer software like KaZaa was a violation of the copyright of the owner of the works. The RIAA was not going after people downloading music, or even people who had sent a file to someone else, but rather the set of people who had shared directories with files that looked like music and were available for public perusal.
This was problematic because the copyright statute doesn't actually say anything about "making available." The right to control who distributes one's work is one of the rights granted to authors by the statute, but the RIAA had no evidence that file sharers had actually distributed the files, just that they had made them available and that the files could potentially be distributed.
Consequently, the RIAA had to argue that "making available" was actually part of the copyright statute when it wasn't. When the handful of suits (out of 35,000) made it to court, some Judges started to realize this, and through the selfless and amazing work of Ray Beckerman, the legal community slowly turned against the prosecution.
All in all, the RIAA's campaign to sue their own customers was a disaster. CD sales continued to plummet and filesharing's popularity only increased. This is not to mention the public relations catastrophe the industry now faces. Musicians hate being associated with large corporations that the public perceives as evil, and more substantively, musicians have not seen any of the settlement monies the RIAA has been collecting on their behalf.
The RIAA is claiming that the campaign "was successful in raising the public's awareness that file-sharing is illegal" which is demonstrates a gross misunderstanding of the law and technology.
So what is next? The RIAA claims that they will be making deals with ISPs to institute something roughly similar to a 3-strikes and you're out policy against file sharers. The terms and details of these agreements are not flushed out (and will probably never be made available to the public), but on some level, this is a less vicious form of negotiation with the technical realities their industry is facing.
But there is already evidence of ISPs acting in haste to dismantle legal file sharing outfits. TorrentFreak has a story about an open source software tracker having their service revoked by their ISP because they were accused of hosting an illegal torrent of the game Command and Conquer. YouTube already engages in auto-take-downs of videos that are supposedly infringing.
I'm a huge fan of the site LegalTorrents.com and have used it for distributing the uncompressed (~1gb) versions of two Creative Commons videos, A Shared Culture and Media That Matters: A CC Case Study. Because everything on LegalTorrents is free to share (under an appropriate CC or similar public license), it is the perfect counterexample to the RIAA's claim that file sharing is inherently illegal.
Put another way, file sharing in and of itself is not illegal (just as crowbars in and of themselves are not illegal) and sites like LegalTorrents demonstrate this. We should not let the RIAA use the fact that they've abandoned their campaign as a positive cover to privately intimidate ISPs into breaking the Internet.
More importantly, we should keep the pressure on ISPs about preserving network neutrality. When used on the public net and ISPs, deep packet inspection filtering and application layer filtering are violations of network neutrality and if the RIAA is successful in pushing these technologies as "solutions" to the file sharing problem, we are going to have a much larger problem on our hands than 35,000 dispersed lawsuits.