The generation he was a voice for was Generation X. I am a member of that generation, and when we all were teenagers, we didn't have anything that really united us. There wasn't anything like World War II, the Vietnam War, etc. to define our generation. There was the first Gulf War, but there wasn't much controversial about that, and controversy over the Vietnam War defined that generation. World War II was constantly in the news and on people's minds because it involved so many countries and affected the U.S. economy so much, although in a positive way. The first Gulf War did not involve too many countries and was definitely not the defining event of our generation. So, we really were one of the first generations in a long time to have no major political or international controversy or war to define us. Generation "X" was a good title for us because the X indicated we did not really know our place in the current world back then in the early 1990s, nor did we know how our generation would be remembered by history books in the future. The generation of service men and women during World War II were referred to as "The Greatest Generation", but who were we and how would we be remembered?
Kurt Cobain tapped into that ignorance of identity because he himself was a troubled soul, who said in an interview once that he wasn't sure if he belonged in this world. The lyrics of some of Nirvana's most popular songs were full of double meanings and non sequiturs. These lyrics didn't allow you to define the band's real identity, and likewise Generation X members could relate to these lyrics since Gen. X could not really define itself either.
In "Smells Like Teen Spirit", Cobain says, "With the lights out, it's less dangerous." You would expect him to say it's more dangerous when the lights are out because then you can't see the dangers around you, but for Gen. X, darkness is less dangerous for us since our true identity is shrouded in "darkness" and difficult to find just like everything else is when there's no light. In the same song, he also says, "I feel stupid and contagious. Here we are now, entertain us." Generation X members feel stupid because they can't put their fingers on who they really are as individuals, nor who they are as a group - as an entire generation. That's why it's contagious, because we all shared that same problem of defining our generation. Like every generation of teenagers, we did want to have a good time and to party. That's why Kurt says, "Here we are now, entertain us.", but as for finding the uniqueness of our generation when it's compared to other generations, we fell short there constantly. There are other examples of lyrics in Nirvana songs that indicated he was speaking for a generation without a real identity.
Also, because Kurt and his band Nirvana were the very first ones to go mainstream with this brand new rock form known as Grunge, he was at least giving Generation X one unique aspect to their generation, a new form of music that evoked different emotions than all other forms of music that had come before it. This also made Kurt the voice of our generation since his music and brand of rock became very popular for the first time during our generation's early teenage, formative years.