Brief Summary
The Set-Up
Even though employment discrimination was outlawed by the Civil Rights Act of 1964, not everyone got the memo. Because…they didn't want to. The Black sanitation workers of Memphis, tired of fighting an unyielding city administration for better working conditions, decided to call in the cavalry: Martin Luther King, Jr. and the Southern Christian Leadership Conference (SCLC).
The Text
This speech covers a lot of ground, but it can be divided pretty straightforwardly into six main parts.
1. First, Dr. King talks bout how human freedom has been a long time coming, but the world has finally reached the point where people everywhere are demanding that their rights be respected. Oppressors are recognizing that that might just be a good idea.
2. Then he talks about the matter at hand, the Memphis sanitation strike, urging the audience to remain unified in their commitment to nonviolent activism and reminding them that nonviolence achieved great victories for the Civil Rights Movement in the recent past. (Example: civil rights actually exist now.)
3. Next is a little interlude: King takes a moment to praise the religious leaders in the room for recognizing that this world is important, not just the next. He's glad they're helping the needy in material ways as well as spiritual. Because it would be nice to actually want to stick around for a while—who knows if they have cheese in heaven? Plus, everybody needs shoes.
4. Speaking of which, King asks the audience to do more than just march. He wants them to combine their economic power and only give money to companies that implement fair labor practices, especially toward African American workers. He also wants them to bolster Black businesses, because that's a win-win: more money, fewer problems.
5. To drive all these directives home, Dr. King reminds his audience of the Good Samaritan, who's basically that dude who always does everything right and everyone thinks he's such a nice guy and you sort of don't like him because his very existence makes you feel bad for not measuring up but you can't quite not like him because, to be fair, he actually is a really great guy. So King tells everyone to put on their Samaritan pants and nice up: it's everyone's responsibility to help everyone else. Even when it hurts.
6. Finally, King reflects on his work in the Civil Rights Movement and ponders his own mortality, speculating that others might have to go on without him. He's right. Still, he remains confident that justice will prevail.
TL;DR
Other people matter, so let's all behave accordingly. It'll make the world a better place, promise.