What does Scout see clearly for the first time when she reads Underwood's claim that Tom's death was senseless killing?Chapter 23

The quote you are referring to is at the end of Chapter 25 -

""Mr. Underwood's meaning became clear: Atticus had used every tool available to free men to save Tom Robinson, but in the secret courts of men's hearts, Atticus had no case.  Tom was a dead man the minute Mayella Ewell opened her mouth and screamed."

Scout understands how deeply the roots of prejudice run.  She sees that you can change the laws and courts and go by all the rules, but unless there is change in the hearts of men, racism and stereotypical behavior will continue.  Tom Robinson outwardly had a fair trial, and Atticus did all he could for him, but because of deeply-ingrained attitudes of inequality in Maycomb, Tom Robinson was doomed from the start.