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).