In To Kill A Mockingbird, Atticus has been tasked with defending Tom Robinson, an innocent man. However, in Maycomb County, innocence is defined in terms of race and, as Tom is a black man, no matter what evidence supports his innocence, he will inevitably be found guilty. Such is the case by chapter 24 when Atticus returns from the prison where he has tried to give Tom some hope of reprieve from his guilty verdict. Tom, however, has given up hope; even telling Atticus that there "ain't no use tryin'." 

Atticus walks in on the Missionary Circle's meeting and it is clear that something is wrong. Atticus tells Calpurnia and Aunt Alexander what has happened to Tom. Tom is dead! He was apparently killed while trying to escape during the exercise period. The guards, having pointed their weapons in the air and fired a few warning shots, then shot him no less than seventeen times, claiming that he was in "a blind rage." It appears Tom almost made it, going so fast, but the bullets stopped him just as he went over the fence. 

It seems, according to Atticus, that Tom thought his chances of escape over the fence were as good as any "white men's chances;" in other words, as unlikely as it may have seemed that Tom would make it over the fence, it is just as likely as a lawful reprieve (that is, improbable).  Atticus has previously tried to reassure Tom that he has a good chance of being freed but is unable to guarantee it and Tom, having patiently waited and having been subjected to humiliation and a complete travesty of justice, can see no chance, despite his proven innocence, because Maycomb County has effectively made its decision. Atticus's faith in human nature is admirable but Tom's own lack of faith in the system is understandable as it has already let him down so badly. Unfortunately, this situation of mistrust and injustice will continue, almost unnoticed in Maycomb County. 

Atticus's explanation of this is actually in Chapter 24, not in Ch. 25.

He says that the prisoners were out for their exercise period when Tom Robinson just started running for the fence.  He says the guards shot Tom 17 times.

Atticus says that Tom was not willing to hope to get off on appeal.  He says Tom was tired of "white man's chances."

It does seem to make sense that Tom would not trust the legal system.  After all, it cheated him once, why would it do right by him now?  On the other hand, it seems possible that the guards just shot Tom because of what they thought he'd done.