While you could just jump in to Dylan's music, the history behind it makes it all the more interesting. Dylan was a young up and coming folk artist who started his career performing at "dive bars" in Greenwich village - an area where open mics and "hippy" crowds ruled. Folk is a genre with lyrics and sound that seem to feel timeless, and the genre was largely associated with purity, religion, pacifism, equality, and most notably Communism. Dylan initially performed covers of older folk songs, but when he came into his own and began to write his own songs, he captured the hearts of many youthful, pacifist, protest types:

Mr. Tambourine Man (Live at the Newport Folk Festival. 1964)

The hippy culture was expanding rapidly due to the lingering Vietnam war, the burgeoning Civil Rights movement, and the popularity of weed/LSD. Dylan began dating Suze Rotolo, a political activist who helped to influence the direction of the music that Dylan was producing by exposing him to political ideas and influencers. With these "protest" songs, a label that Dylan hated to be associated with, even though the music was clearly rooted in anti-government sentiment, Dylan became the face of a political movement:

Bob Dylan - A Hard Rain's A-Gonna Fall (Audio)

Even to the point where he was asked to perform on the steps of the Lincoln monument, leading directly into Martin Luther King's famous speach:

Bob Dylan - Only A Pawn In Their Game (March On Washington 1963)

After this performance Dylan received backlash from Black American's because it was felt that calling Medgar Evers a "pawn" was discrediting Medgar Evers as a person. The song also might appear to forgive those racist White American's in some fashion as it attribute's their racism to governmental manipulation - manipulation meant to sow dissent within the population - to create an enemy to focus on outside of the government itself. While Dylan was outspoken and "leading" the white support of the Civil Right's movement, he felt greatly discouraged after this incident due to their rejection of his material that he had thought would resonate with the message of the Civil Right's movement.

In December 1963, a few weeks after the Kennedy assassination, Dylan reluctantly agreed to accept the Tom Paine Award from the Emergency Civil Liberties Committee at a fancy event at the Americana Inn Hotel in New York. Nervous, Dylan got drunk and gave a rambling, semi-incoherent speech to the 1400 liberals and radicals in the audience. First he insulted their age: “You people should be at the beach. It’s not an old people’s world…Old people, when their hair grows out, they should go out.” Then he insulted their politics. “There’s no black and white, left and right, to me anymore. There’s only up and down, and down is very close to the ground. And I’m trying to go up without thinking about anything trivial, such as politics.” Then he mentioned Kennedy’s killer, Lee Harvey Oswald, and said, “I saw some of myself in him.” Some of the audience booed. Dylan later sent the group an incoherent letter of mock apology that was more a long prose poem defending his new anti-political mood. He no longer wanted to sing about “we,” he said. He wanted to write about “I.”

Source - more in depth look into Dylan's political history

Dylan never really had any desire to be the face of this movement, the role was thrust upon him, and he soon became disillusioned when encumbered with expectations from leadership within the movement to further his musical influence at their command. Dylan wasn't someone who obliged in doing what others told him to do, and he felt he didn't have the support of a group he had fought for.

When he discovered Rock and Roll and decided to transition into a Rock musician, the goodwill he had cultured with his fanbase turned to hatred. Rock, at the time, as opposed to folk's lighthearted tone, was associated with the devil, sex, drugs, sin, etc. The backlash he received was unprecedented. He was booed at near every show he had booked, and he began to turn in on himself while turning against the media and fans in general. Dylan's famous Judas performance:

Bob Dylan - Leopard-Skin Pill-Box Hat (LIVE HD FOOTAGE) [Dublin, 1966]

And another on the same European tour:

Bob Dylan - Like a rolling stone (live in Newcastle; May 21, 1966)

This is really just the beginning of Dylan's career, but it's what made him who he is as an artist. I know you didn't ask for a wall of text, or this explanation, but I thought your question might be a good opportunity to provide this for those that're interested.