How does "Letter from Birmingham City Jail" combat racism?

In Martin Luther King Jr.'s "Letter from Birmingham City Jail," King combats racism by championing the cause of nonviolent resistance, equating different groups of people with one another rather than separating them, and calling people to action who have remained silent out of a desire for social order.
When King was imprisoned in a jail in Birmingham, Alabama, he wrote this letter in response to critics who said that issues of racism and segregation should be fought in the courts rather than in the streets. These critics were specifically eight white clergymen who wrote an article in the local newspaper titled "A Call for Unity." King used his platform as both a minister and Civil Rights leader to champion the cause of nonviolent resistance.
To understand King's "Letter from Birmingham City Jail," one first has to understand exactly what nonviolent resistance was to King.
Nonviolent resistance is, according to King, "a courageous confrontation of evil by the power of love." It can also be defined as protesting through nonviolent means like symbolic protests, disobedience to unjust laws, economic noncooperation, and other forms without violence. King became interested in it after reading Thoreau's "Essay on Civil Disobedience" while he was in college.
When people criticized the methods that people in the Civil Rights movement used to protest segregation and racism, they meant that those problems would be taken care of within the law. They believed it was unnecessary to do things like sit in at lunch counters, refuse to give up a seat on the bus, or not cooperate with other unjust laws. They felt that it widened the divide and created more chaos.
King disagreed. In his letter, he argued that it was not only necessary but also right to practice nonviolent resistance. 
For King, there were six parts that defined nonviolent resistance. They are:
A person can resist evil without using violence.
Nonviolent resistance wants to gain friendship and understanding from their opponents.
Evil as a concept—not the people doing the evil acts—was what they were resisting.
Nonviolent protesters had to be willing to suffer without retaliation; King believed suffering could be redemptive.
Protesters had to avoid both internal and external violence. It was important for protesters to refuse to hate their opponents.
Protesters had to have faith in the future and believe that the universe is on the side of justice. 
So that's the concept he was championing in his letter.
One way that King combats racism is to make it clear that he sees all people as the same. He states that he isn't there to create trouble but rather to fight for a better world for everyone. He's standing up to the idea of himself and his cohorts as outsiders who came to create dissension when he writes,

Moreover, I am cognizant of the interrelatedness of all communities and states. I cannot sit idly by in Atlanta and not be concerned about what happens in Birmingham. Injustice anywhere is a threat to justice everywhere. We are caught in an inescapable network of mutuality, tied in a single garment of destiny. Whatever affects one directly, affects all indirectly. 

Next, King makes a distinction between just and unjust laws. This helps combat racism by making the unfair and negative intentions of the laws he and his group were protesting clear. He says that just laws uphold human dignity, while unjust laws degrade it. He says that they hurt both the oppressed and the oppressors. Again, he is grouping people into a single group instead of letting the boundaries of race remain.
As an example, King discusses segregation. He says that the law isn't democratic because Alabama denies black people the right to participate in the democracy. He argues that laws are created specifically to uphold segregation—like the law that banned parading without a permit, for which he was jailed. King says that he is willing to pay the price for breaking that law.
He also combats racism in his letter by using examples to show his meanings and get his point across. For example, he compares segregation and its dependence on law to the laws of Nazi Germany that allowed for Jewish persecution. He explains that those were laws he, too, would have been willing to break.
Much of his letter from that point is a call to action for church people and white moderates. King explains:

I have almost reached the regrettable conclusion that the Negro's great stumbling block in his stride toward freedom is not the White Citizen's Counciler or the Ku Klux Klanner, but the white moderate, who is more devoted to "order" than to justice; who prefers a negative peace which is the absence of tension to a positive peace which is the presence of justice; who constantly says: "I agree with you in the goal you seek, but I cannot agree with your methods of direct action"; who paternalistically believes he can set the timetable for another man's freedom; who lives by a mythical concept of time and who constantly advises the Negro to wait for a "more convenient season." Shallow understanding from people of good will is more frustrating than absolute misunderstanding from people of ill will. Lukewarm acceptance is much more bewildering than outright rejection.

He goes on to say that the entire purpose of law and order is to create justice and uphold it. Therefore, their failure to do so is dangerous and damaging. He feels that white moderates and people of the church fail to realize this—but still can. He says that they can expose injustice to "the light of human conscience and the air of national opinion." 
Throughout the letter, King keeps his tone polite and makes a point to appeal to both logic and emotion, citing both historical precedents and the feelings of his young daughter, who cannot go places because she's black. In this way, too, he makes a connection with readers who might otherwise be disinclined to even consider his words.
The letter was signed: "Yours for the cause of peace and brotherhood, Martin Luther King, Jr."