The Fairy Dance arc gets a lot of criticism for various reasons- from the Happy Ending Override that forces Kirito to save Asuna again to how Kirito ultimately defeats Sugou/Oberon. While it does have its flaws, it's also very important to Kirito's character arc. By showing Kirito returning to the real world, while still having unfinished business in the virtual one, the arc displays how his time in SAO has changed his beliefs on the real and virtual worlds, as well as the boundaries between both.
To begin with, rescuing Asuna isn't Kirito's only goal in this arc. He also has to complete his rehab, return to a school designed specifically for former SAO players like him, and reconcile with his adoptive sister, Suguha(which will come up in a few paragraphs). The transition from the game of death to the real world is not seamless, and Kirito's correct when he says that he feels as though SAO isn't completely behind him.
It's worth discussing Kirito's relationships with other players. Like many people who play online games, he'd previously thought of the player and their character as two separate entities, with the latter being a false persona that the former assumed while online. All that changes when he's thrust into SAO, a game in which people live inside the game world, wearing the same faces that they do in real life and dying when their characters do. When PKers are essentially the same as murderers, it thus makes sense that Kirito would come to realize that if you behave badly in a video game, under the guise of anonymity, you might just be a bad person at heart. With that in mind, he rejects Titan's Hand's belief that Akihiko Kayaba has absolved them of all responsibility for their actions, and helps Leafa out, simply because it's the right thing to do.
Of course, ALO is a more standard VRMMO, and while Kirito's racing to rescue Asuna, most of the other players- including Leafa, Kirito's companion- think of it as just a game. As such, Leafa, depressed and disillusioned with ALO, suggests that Kirito might be better off betraying her and teaming up with her more powerful enemies, the Salamanders. Kirito, despite knowing of the stakes, refuses, simply because he believes that "The player and the character are one and the same," and he doesn't want to turn on a friend for personal gain. It's the more difficult route, albeit one that pays off in the end, but this goes a long way toward showing Kirito's strong moral integrity.
Speaking of Leafa (aka Suguha), while she often gets criticized for her incestuous crush on her cousin/adoptive brother, she actually never wanted to feel that way. She knew that while relationships between first cousins are accepted in Japan, relationships between adoptive siblings are not, and tries to get over her crush on Kazuto... by falling for Kirito, who is Kazuto. Unfortunately, Suguha can't talk to Kazuto about this, and because of rules against talking about real life matters online (a policy that apparently serves to enforce the divide between player and character) Leafa can't tell Kirito, either. It takes until the aftermath of Kirito's failed first assault on the World Tree, and mentioning that he has to save Asuna (shortly after Suguha first laid eyes on the comatose Asuna in the hospital), for Suguha to put the pieces together- from what Kirito just said to Kazuto imitating Kirito's way of sheathing his sword- and realize Kirito and Kazuto are the same person.
Naturally, in the fallout, Kirito is forced to once again return to the realization that players and characters are one and the same, and another, more personal one- even if he's actually adopted, it doesn't change the relationship he's had with his parents and sister. While Kazuto had returned home with the intention of repairing his relationship with Suguha (another result of The Black Swordsman arc), only after thinking things through and realizing that he had to see Leafa as she was (combined with Leafa having a similar epiphany) are the two siblings able to patch things up.
Finally, there's the villain, Sugou, who's both incredibly monstrous and almost laughably incompetent, but nevertheless very dangerous. In the real world, his only hope of gaining power is to trick Asuna's father into giving him his daughter's hand in marriage, rather than advancing himself through hard work or intelligence, so he puts up a polite front to ingratiate himself to Shouzou, while treating Kirito with disdain once Shouzou has left the room. In the virtual world, he resorts to mind control and abuse of his administrator privileges, and while he pretends to be Oberon, King of the Fairies, it's little more than showing off his true colors as a lecherous and sadistic tyrant. When he attacks Kirito outside the hospital, it's clear that Sugou's lost all reason and the desire to keep up his facade. Without the resources at his disposal, Sugou would be a relatively pathetic, if despicable, individual, but with the means to make his twisted schemes reality, Sugou becomes a significant threat to others' well-being.
So how does Kirito deal with Sugou? In the game, after gaining Kayaba's administrator powers, Kirito turns the tables on Oberon and kills his character, to secure his escape with Asuna and get some payback in the process. In the real world, however, Kazuto disarms Sugou, then knocks him out, ties him up and leaves him for the police. One difference between the two worlds is that the real one is governed by the rule of law, so Kazuto decides Sugou deserves to be judged for his crimes. When you consider that Kirito not only defeated and (seemingly) killed Kayaba tow in the game, but also had to kill three members of Laughing Coffin in the past (which he regrets), this difference between the real world and the kill or be killed nature of online becomes apparent, but in both cases, Kirito does what he believes to be right.
This theme also comes up in other arcs. Shino Asada, co-protagonist of the Phantom Bullet Arc, hopes that by becoming her character, Sinon, she can free herself of the trauma she suffered as a young girl, but realizes that Shino and Sinon are two halves of the same person, and only takes the first steps toward healing by making peace with the incident as Shino. By comparison, the arc's antagonists have a very unhealthy relationship with virtual reality- one uses it as an escape from his real life and goes so far as to plot a Murder-Suicide with Shino when he fails to become the strongest in GGO, while the other two are former Laughing Coffin members who continue their murderous ways in the real world. Once again, GGO returns to the theme of the real world and virtual reality, thus following up on Fairy Dance.
Despite primarily being set in a VRMMO, Sword Art Online has surprisingly strong roots in the real world, and this is perhaps emphasized most in the first arc after Kirito's escape from the game, where he divides his time between playing as Kirito in Alfheim and reclaiming his life as Kazuto Kirigaya in the real world. Just as Kirito has changed a great deal during the past two years, so has Kazuto, since the two are one and the same, and you see the changes at play in this arc. In that sense, for all its shortcomings, the Fairy Dance arc fits with series-wide themes in Sword Art Online, and while some viewers would like to forget it existed, it still serves an important purpose in the narrative.