DIll needs to leave the court room during Mr. Gilmer's cross examination of Tom because Mr. Gilmer's treatment of Tom makes Dill sick. Mr. Gilmer is calling Tom "boy" and being condescending and Dill understands that isn't right. Atticus was kind to Mr. Gilmer's witnesses, so there's no reason Mr. Gilmer shouldn't be nice to Tom.
Dolphus Raymond is commenting on the fact that children can see injustices far better than adults--or are more sensitive to them than adults. Dill can understand that Mr. Gilmer's treatment of Tom is unjust, but most adults in the court room will turn a blind eye to it because Tom is black. This foreshadows Jem's reaction to the verdict--Jem can understand how wrong it is, and can't understand how adults--who are supposed to be so smart--could come to that verdict. Kids aren't overly sensitive--they just don't see the same differences that adults do,