Dill shows his sensitivity in this chapter because of the way that Mr. Gilmer harangues and challenges Tom Robinson as he is cross-questioned at court. Mr. Gilmer is particularly insulting in the way that he refers to Tom Robinson as "boy" and also in his tone and what he seeks to insinuate in his words. Note what Scout says about how Dill responds to this:
For some reason, Dill had started crying and couldn't stop; quietly at first, then his sobs were heard by several people in the balcony.
Dill later explains to Scout that it was Mr. Gilmer and the way he spoke to Tom that made him start crying, as it was "wrong," and it "just makes me sick" for somebody to talk to somebody else--negro or otherwise--in that kind of insultng and patronising tone of voice. Dill therefore shows his sensitivity at the way that he is able to get upset over the terrible way Tom Robinson is treated and he recognises that there is something supremely wrong with how Mr. Gilmer is talking to Tom and treating him. He instinctively recognises that "it ain't right," and this shows him to be far more sensitive than other characters who do not see anything wrong with how Mr. Gilmer is talking to Tom.