In Chapter 15, it is after a week of Dill being in town with that the events surrounding the trial of Tom Robinson really pick up. So, the lack of peace has nothing to do with Dill himself; he just happened to show up before things really got intense. It is a week after Dill arrives that Heck Tate and a group of men come to the Finch house and try to convince Atticus that Tom shouldn't stay in the Maycomb jail that weekend because they are worried that there might be a riot or some sort of violence. Atticus refuses, and then goes and plants himself outside of the jailhouse, just to keep an eye on things. An angry mob does in fact show up, and it could have gotten really scary if Scout hadn't unintentionally broken the tension by talking to Mr. Cunningham.
So, Dill's arrival just happens to be right before the trial, which Scout said was "a nightmare".
At the beginning of Chapter 15, Dill is allowed to stay with his aunt in Maycomb and Scout mentions that they had a week of peace before a "nightmare" was upon them. Scout's "nightmare" foreshadows the dangerous situation that takes place outside of the Maycomb jailhouse later in the chapter. As was mentioned in the previous post, Atticus saves Tom Robinson from a lynch mob that shows up at the jail. The children end up finding Atticus and narrowly avoid a tragedy when Mr. Cunningham realizes the error in his judgment after acknowledging Scout's presence. As Tom Robinson's trial draws nearer, certain members of the community of Maycomb becomes increasingly hostile, and the Finch family is subjected to criticism and danger. Even following Tom's wrongful conviction, the Finch family is in danger because Bob Ewell seeks retribution for Atticus "ruining" his reputation.