Atticus speaks these words in the presence of Calpurnia, Aunt Alexandra, Scout, and Miss Maudie.
By the time these words are spoken, Tom has already been shot dead. By all indications, Tom was caught trying to escape from prison. The guards had fired off a few warning shots, but Tom had ignored those. Eventually, Tom's body was pumped with seventeen bullets as he tried to make his way over the fence. The guards maintained that Tom would have managed to escape if he had had two good arms to propel him over the fence. Essentially, Tom decided that he would rather take a bold risk than rely on Atticus' promise of an appeal.
Atticus' words are significant in that he is expressing an ugly truth about the Jim Crow era: even if Tom receives a new trial, his chances of acquittal are little to none. The court of public opinion has already condemned him. No matter what he does, Tom's guilty verdict will haunt him for the rest of his life in Maycomb, and he will always be the subject of malicious gossip. His life and the lives of his family will also be at risk. The only conceivable way to survive would be for him to try to escape and to get his family out of Maycomb.
Atticus' words about Tom's distrust of "white men's chances" prove prescient. Immediately after Tom's death, the community gossips for two days about the events surrounding his failed escape. However, true to habit, the community soon moves on to other topics of interest. The general consensus is that "Tom's death was typical" and additionally, there is little remorse for his death.
Typical of a nigger to cut and run. Typical of a nigger’s mentality to have no plan, no thought for the future, just run blind first chance he saw. Funny thing, Atticus Finch might’ve got him off scot free, but wait—? Hell no. You know how they are. Easy come, easy go. Just shows you, that Robinson boy was legally married, they say he kept himself clean, went to church and all that, but when it comes down to the line the veneer’s mighty thin. Nigger always comes out in ‘em.