Setting the scene: the song was released in 1963 and included in the album The Freewhelin' Bob Dylan. All of Dylan's "most famous political songs were written [...] between January 1962 and October 1963" and "those [...] fixed him in the popular imagination" as a protest songwriter (Lynskey, 2010, 67).
The historical context in which the song was released is essential to understand its meaning: during the 1950s the civil rights movement against racial segregation started to gain momentum, reaching its highest point precisely in 1963. Indeed, in that year:
* Desegregation protests spread throughout the Southern states over more than 100 cities. The most famous was the March on Washington: it was organized by the "Big Six" of the civil rights movement (Martin Luther King Jr., James Farmer, J. Lewis, P. Randolph, R. Wilkins, W. Young), attended by more than 250.000 protesters (including Bob Dylan). It was in this occasion that the renowned "I have a dream" speech took place.
* On June 12, President Kennedy announced he would present a civil rights bill to Congress, which was eventually passed the following year under the name of the Civil Rights Act.
https://www.britannica.com/event/American-civil-rights-movement https://www.britannica.com/event/American-civil-rights-movement
It is important to bear in mind that protests were not only motivated by the insufferable racial segregation, but they were also anti-war oriented: the Vietnam war (1955-1975), which had been raging for ten years by the mid-1960s, was felt as an unnecessary conflict, especially by students and young people, who were the ones recruited in the U.S. army. Indeed, "the average age of an American soldier in Vietnam was 19" (https://www.bbc.co.uk/bitesize/guides/z6dk8hv/revision/4 ).