You would have to know a bit about how America's social systems among teens worked in middle-class towns.
The "socs" were from "good" families who had money and status in the town. They wore it in their clothes, cars, and parties. They were the kids who had it all.
The "greasers" were from working class neighborhoods, broken homes, or other lower status stations in community life. The greasers identified themselves by their hair, leather jackets, etc. They did not have money and position, but they had their pride.
The main external conflicts in the story were an ongoing conflict over "soc" girls liking "greaser" boys and social status in the community.
The story is not really very different from Romeo and Juliet or Westside Story.
Two drunk socs see Ponyboy leaving a movie theater with soc girls. They stop their car jump out and begin to beat him and drown him in a water fountain. Ponyboy is saved by his brothers and other greaser friends. In the process, Johnny recognizes one of the drunk socs as one who had previously beaten him up. Johnny stabs this "soc" boy to death with his knife.
Ponyboy and Johnny have to leave town to hide out while the murder investigation is sorted out. When Ponyboy and Johnny dyed their hair blond and cut it short, they felt they had lost themselves. They did not have all the nice cars and clothes that the socs had, but they had their "look" and now that was gone.
The "big fight" is basically to settle the score. Johnny was beaten up, Ponyboy was nearly drowned. Both Johnny and Ponyboy ran away and hid due to the knifing that killed Bob. Johnny and Ponyboy end up becoming heroes by saving the children from a burning church where they were hiding.
While Johnny is in the hospital, the "rumble" occurs to establish who is the real wininer. The greasers end up winning the fight. The socs are told to stay on their side of town and leave the greasers alone.
They fought in the rumble because they wanted to get left alone by the Socs.
They are fighting this rumble because, the greasers want the socs to stay on there side and leave the greasers alone! They wanna be pround of themselves
they fighrt in the big fight because, if you kill one of their gang members then you'll have to fight just to get it even with them.