I'll try to answer this as short as possible.
1. Because Axl Rose has anger issues.
2. Because occasionally Axl shows late for shows (or at least he did back in the day).
3. Because Axl is known to leave in the middle of a show when "fans" are disrespecting him. Once somebody threw a bottle at Axl on stage, and that made him leave.
4. Because once he started a riot at St. Louis. A fan was filming with a camera from the front row and Axl literally jumped offstage to take the camera away from him. Cameras were not allowed, but the fan brought it anyways and Axl saw that. Afterwards Axl said "Thanks for the lame-a$$ security, I'm going home" before leaving the concert. Then there was a historic riot and Axl Rose got sued.
5. Because he is one of the reasons the original Guns N' Roses lineup split. Axl Rose DID NOT fire them (the only member he actually fired was Matt Sorum in 1997), they left on their own will. Steven Adler was fired by the whole band in 1990 because of his drug problems.
6. Because Axl and Slash have beef ever since and don't talk to each other.
7. Because Axl refused to be inducted in the Rock N' Roll Hall of Fame (which really is a bad joke) in 2012.
8. Because it took him 15 years to release a new album with "the new" Guns'. Which wasn't a bad album, but a lot of people hate it (maybe because it has a whole new lineup).
9. Because he seemengly starts fights for no reason.
10. Because Kurt Cobain and Axl Rose had a beef. Courtney Love asked Axl if he could be the godfather of their baby, but Axl took it the wrong way, as an insult, and called her a "*****". He started threating Kurt and Cortney, which led to a beef between Axl Rose and Kurt Cobain. Nirvana band members and Guns' band members were (and still are) okay with each other, except for Axl.
11. Because him and Metallica have a beef. Axl once said in 2008 that his favorite cartoon characters were "Metallica and Slash".
12. Because he's often misunderstood.
On that note, Axl Rose is really a nice guy when he's not pissed off. He has anger issues, but who doesn't? If you were in his shoes and someone threw a bottle at you in the middle of a concert, would you not leave too? I surely would.
He had the full right to take the camera from that fan in St. Louis as cameras were FORBIDDEN and the SECURITY should've done something about it. They did not and Axl left. I'm not saying that's cool, but he had the right to do as the fan violated the rule.
Axl Rose did NOT fire his band members. Steven Adler was fired in 1990 by the whole band because of his drug problems. Izzy Stradlin left on his own in 1991, they are still in a good relationship today. Slash left in 1996 after a fight with Axl Rose. Matt Sorum was the only member that Axl actually fired in 1997, because he defended Slash. And finally Duff McKagan left on his own in 1998, which led to Axl forming a whole new lineup (alongside Dizzy Reed who's in to this day). Axl seemingly hates Slash because he left Guns N' Roses.
Some people hate on Axl Rose for not wanting to join the Rock N' Roll Hall of Fame. I think he did good by not wanting to have anything to do with it, since RNR Hall of Fame is one bad joke. I mean, Madonna is in it after all. And recently Public Enemy was inducted, which is hip-hop. But on that note, Deep Purple is not in. How is Madonna, a pop singer better than Purple? How is Public Enemy, a hip-hop band better than Purple?
Note that Paul Stanley and Gene Simmons of KISS also thinks of the RNR Hall of Fame as a joke and refused to get inducted, yet everyone seems to be is okay with that, but people still hate on Axl Rose for declining.
Don't be surprised if one day the likes of Britney Spears, Ke$ha, Justin Bieber, Miley Cyrus, Beyonce, Jay-Z, Nicki Minaj, Chris Brown, Rihanna, LMFAO, Skrillex or PSY get inducted as well. They all have two things in common - they're not one bit rock & roll and they all suck. I mean, they did it by inducting Madonna after all...
It took Axl Rose 15 years to release a new album since the last Guns' album in 1993. With a whole new lineup (apart from him and Dizzy), Guns N' Roses released "Chinese Democracy" in 2008. A lot of fans hate this album, but then again, a lot of fans love it. I think it's a really good album, just not as good as the previous Guns' albums with Slash. But a good album nonetheless. You'll either like it or hate it.
To sum it all, Axl Rose is awesome no matter what anyone says about him =) He works hard to deliver the best shows possible, yet people will always find a way to disrespect him. He has his flaws like anyone else. I really doubt he's as bad as people make him up to be, just misunderstood. Love him or hate him, you have to at least show some respect.