"Simply because we were licked a hundred years before we started is no reason for us not to try to wish." How is this statement relevant to the...

Scout asks Atticus if they are going to win the case. Atticus responds honestly that he does not think they will win. Scout asks why he would even take the case. This is when Atticus says: 

Simply because we were licked a hundred years before we started is no reason for us not to try to win. 

Scout says this sounds like the thinking of Cousin Ike Finch who said he would fight for the Confederacy just the same even though they lost the war. This reference to the Civil War is significant to Atticus's previous statement. The novel takes place in the 1930s. One hundred years prior would have been the 1830s, about 30 years before the South was "licked" in the Civil War. 

The South fought to uphold slavery and it was a losing battle but something the South was going to fight anyway. Atticus notes that the case of Tom Robinson is different. Clearly, the cause is more righteous with Tom Robinson than trying to preserve slavery. Even though he knows he won't win the case, he still intends to do his best. There is the notion that it may be many more years (perhaps not 100) before a black man can get a fair trial in the South. But that is no reason not to try. Atticus explains why he must take the case: 

The main one is, if I didn’t I couldn’t hold up my head in town, I couldn’t represent this county in the legislature, I couldn’t even tell you or Jem not to do something again. 

This statement is relevant for Atticus because he tries to be consistent with his words and actions. He tries to be the same fair and reasonable man in his home as he is in the courtroom and on the street. So, he has to take the case because it's the right thing to do. This decision to take a controversial case will affect the children as well. They will be mocked at school and they will eventually face danger because of it. Even though Atticus loses the trial, Bob Ewell will continue to hold a grudge simply because Atticus defended Tom. So, the simple act of taking the case will affect all of the main characters and will culminate in the climax of the novel.