There are a number of ways that Bob Dylan influenced American music (and English music). His influence is often associated with counter-culture of the 1950s and 1960s in the context of socially conscious folk music, but his counter-cultural influences can be seen to continue into the late 1960s in electric-guitar oriented rock and roll music.
Early in his career, Dylan was known for two things, it seems. He was a traditionalist in the sense that he was an aficionado of traditional American musical styles and musicians. He was famously enamored with the music and the persona of Woodie Guthrie, for example, and explicitly attempted to carry on the populist ethos that Guthrie represented.
Populism was counter-cultural in a political context during a period of American history where authority was arguably more patriarchal, centralized and vaunted than it is today. It was a time of the McCarthy trials, of post-war opulence and wide-spread expectations that if one conformed within the clearly articulated social matrix, one would get ahead and be successful.
These views of conformity and acceptance of authority were challenged by non-conformists who reacted against the mechanization of American life (by championing folk traditions from an earlier era) and who pointed very vocally to the unequal treatment of American citizens in terms of race. Conforming to a system of materialism and prejudice was anathema to socially progressive musicians of the day like Bob Dylan and Joan Baez.
Dylan expressed views on these issues rather directly and became a central voice for a social consciousness coming from Berkeley and Greenwich Village. Some of the songs Dylan wrote and performed became known as protest songs (a label he would later resist and evade). The political content of these songs was timely and clearly stated.
How many roads must a man walk down
Before you call him a man?
How many seas must a white dove sail
Before she sleeps in the sand?
Yes, and how many times must the cannon balls fly
Before they're forever banned?
Outside of his traditionalism and his politics, Dylan was known for bringing a literary quality to his music. This facet of his music connected him with the Beat poets of his generation and helped to inspire song-writers for decades to come, some of whom may have taken part in counter-cultural movements of their own day.
"Combining forms borrowed from folk ballad verse, blues, country and western, and gospel music and techniques gained from French symbolists and beat poets, Dylan revitalized the popular song and inspired other musicians to follow his lead in self-expression" (eNotes).
Famously, Bob Dylan influenced the Beatles to explore electronic rock and roll and introduced that band to a new psychedelic mindset that would characterize its music for years to come (and the music of countless other bands as well). Though Dylan was given a hard time for "going electric" he was successful in paving the way for others to be accepted with that format (e.g., Jimi Hendrix and even Miles Davis).
Dylan cannot be credited with creating electric-guitar rock and roll. He can, however, be given credit for playing a part in pushing the boundaries of mainstream music. Plugging in, Dylan made a bold statement about the extent to which he would allow himself to be defined by others -- including his fans. He was no longer associating himself with American traditional music (though later he would return to it). Instead, he was exploring new territory and putting away politics as well.
Dylan's contribution to American culture is probably best seen as one of articulation and interpretation rather than leadership. He was never a civil rights leader, as he has said himself. He is a writer and a performer, inspiring for his political stands and for the quality of his art. He remains a uniquely preeminent American artist, probably best seen in a category of his own.
He was the first to do many things and perhaps the best at doing some things (like writing protest songs, if we still want to call them that).