That Dill is sensitive has already been demonstrated in earlier chapters of To Kill a Mockingbird. For, he runs away from a home in which his mother is distant. Yet, it is ironic that he becomes so upset at the prosecutor's badgering of Tom Robinson in light of his creating so much mischief against Boo Radley.
Perhaps, Boo Radley seems unreal to Dill, but as he sits in the courtroom and perceives the obvious cruelty of Mr. Gilmer to Tom Robinson, with his child's mind that sees more clearly that the jaded adult minds, Dill feels the sting of bias. An an underdog himself, he is sensitive to this cruelty of Mr. Gilmer, and is physically shaken by it. His reactions of becoming sick because as Mr. Dolphus Raymond remarks, "You aren't thin-hided, it just makes you sick," foreshadows the remark of Atticus Finch who says after the trial that only children cry about the "secret courts of men's hearts."
The short answer here is that it gets him really upset. In fact, he gets so upset that he actually has to leave the court. He's just about physically sick. You can find this right at the end of Chapter 19.
What appears to bother Dill the most is the total lack of respect that Mr. Gilmer has for Tom Robinson. As Mr. Gilmer keeps calling Tom "boy," Dill gets more and more upset. Finally, he starts crying and has to leave the courtroom.
This shows just how unfair and hostile Mr. Gilmer is being to Tom.
Dill is shocked by the harshness and cruel methods which Mr. Gilmer uses to cross-examine Tom Robinson. He erupts in tears, unable to contain his horror and is escorted by Scout outside the courthouse.
As a young boy and child, Dill is very pure: he has not yet been stained by the injustices of racism and does not yet understand the harsh truth - that the word of a white will always preside over an African-American's. Adults are not fazed by these cruelties and view it as more of a habitual occurence because they have already lost this pureness: because they have already experienced such cruel lessons and learned the unfair boundary racism creates.