"There's something in our world that makes men lose their heads -- they couldn't be fair if they tried. In our courts, when it's a white man's word against a black man's, the white man always wins. They're ugly, but those are the facts of life." (pg 220)
What Atticus is predicting in your quote is that the black people were only going to take it for so long, and then they were going to rebel. He tells his children at the end of your quote,
"I hope it's not in you children's time" (pg 221)
Unfortunately, it all came to a head in the children's time. The following historical events happened starting in 1963.
1963: Birmingham, Alabama (Remember Maycomb is in Alabama): The first Civil Rights marches took place. This is considered the pivotal point of the Civil Rights movement. Rev. Martin Luther King was from Birmingham, Alabama.
1964: Harlem Riots: This was a four day riot after an off-duty police officer shot an African-American teenager in Manhatten. The officer said that the young man pulled a knife on him while the boys were harassing a building owner.
1965: Watts Riots: These riots lasted from August 11-15 and were in a suburb of Los Angeles. A 27-year-old African-American man was stopped by a white motorcycle officer for possible drunken driving. When they tried to arrest him, a fight broke out. His brother was a passenger in the car, and his mother lived nearby. They fought the officers also and were arrested. A crowd grew and continued to grow until the police chief had to call in the National Guard. $40 million dollars of damage was done in this riot and over 300 people died.
1966: Hough Riot - This was located in a suburb of Cleveland, Ohio. It started as an argument in a white-owned bar that was located in in a black neighborhood. The success of the automotive industry was hitting Cleveland, and everyone was benefiting except the African-Americans. They still remained poverty-ridden. The article I have linked below states,
"...the triviality of the incident suggests strong currents of persecution and animosity that were desperately seeking an outlet."