This is called 구개음화 (palatalization), and it's one of several assimilation rules in Korean. It occurs when ㄷ or ㅌ is in the 받침, and is followed by a syllable beginning with 이:
같- + 이 = 같이 [가치]
굳- + 이 = 굳이 [구지]
In the example you've given, the ㄷ + ㅎ combination produce [ㅌ] sound, but since it is followed by the 이 sound, palatalization occurs, resulting in [치] pronunciation.
(Note that this rule only applies when the ㄷ/ㅌ is in the 받침 (and thus ending the morpheme). Historically, this sound change occurred across the board, but where it occurred in the middle of the word, the orthography was updated:
텬디 -> 천지
Now, exceptions like 디디다 remain; this should be pronounced as [디디다], without palatalization, because the ㄷ is not in the 받침).