In my opinion, Guns 'N' Roses, along with latter-day U2, Madonna, Michael Jackson, and Depeche Mode, represent the changing of the guard from classic to modern rock. U2 helped pave the way for heavily political and emotional music (twee), Madonna codified dance music, Michael Jackson helped popularize and transform R&B into contemporary R&B (a depressing legacy to be sure), and Depeche Mode proved that emotion and rock instrumentation is not necessary for success, giving alternative, trip hop, and (partially) house a pop blueprint.
Guns 'N' Roses is similarly connected to alternative, not necessarily in terms of inspiration, but in terms of popularization. It's a tricky distinction, that I'm going to do my best to justify. Basically, the public was primed for alternative's distorted guitar and huge percussion by Guns 'N' Roses and other bands like them (metal, basically). I've mentioned before the straight line between first-wave metal (Led Zeppelin, Black Sabbath) and certain alternative bands (namely, Soundgarden) so this shouldn't come as a surprise.
The similarities are clearly apparent in terms of the sound/volume. All the bands I mentioned in the last paragraph are characterized by being fucking loud. Turned up to 11 loud. There was variety in their music (acoustic numbers, power ballads, etc) but the loud bits tended to get noticed more, and became a part of all the groups' popular identity. Remember, the obvious/frequent bits are the ones people remember.
Guns 'N' Roses serve as a good example for a turning point mainly due to their place on the musical history timeline ('87, right before Surfer Rosa and Daydream Nation (both from '88)) and their time capsule-esque status. No one really cares about Guns 'N' Roses outside of their first album, Appetite For Destruction, and even if they do it's mainly for certain singles ("Civil War", from Use Your Illusion for example). This is partially because following up on Appetite in a satisfactory manner was virtually impossible (the thing was positively massive, remember) but also because the band began imploding almost instantly. They were working with Axl Rose, one of the biggest assholes in the history of music, whose pride was based on... his... great voice? His lyrics? No, just his image.
That's another bit that directly inspired alternative music. Like Led Zeppelin before them, the sheer excess and grandiosity of Guns 'N' Roses and other, similar bands helped create the cultural context for the stripped down, bare bones approach of grunge and alternative rock. More specifically, it helped create an appetite for such stripped down music (an appetite for destruction you might say. I apologize, that was terrible).
This is not unusual. One can trace musical history as a repeating process of building up and then stripping down. We went from baroque pop to singer-songwriter and folk music. From early metal to punk. From hair metal and R&B to grunge and alternative. Even hip-hop shows this ebb and flow; compare the relatively early major hip-hop albums like Illmatic, Enter The Wu-Tang (36 Chambers), and Reasonable Doubt to recent albums like Speakerboxx/The Love Below, The E.N.D. (FUUUUUUCKING shit), and especially My Beautiful Dark Twisted Fantasy. From classy skeleton to dressed up in a puffy coat and venetian shades. Which you prefer is a matter of taste. I personally prefer the stripped down stuff, but I still love some of the more bombastic pop music out there (Motown, for example). Either one can be done brilliantly (Leonard Cohen vs Queen, for example) or terribly (Jason Mraz vs Rihanna, for example). Neither is inherently better than the other, though later examples of bands/albums in either period tend to look a little worse for wear. Pop music can't stay the same forever, and I'd put my money on a retreat from bombast in the next ten years or so. Especially now that Kanye's topped us off.