1. To provide further explanations/clarifications for a post
For me, chat is there to facilitate another channel of communication. We wouldn't want to limit users to discussions only in the comments.
Looking directly at Stack Overflow, the topics there usually require a little more in depth explanations and answers. Users at different levels of competence will all see the same answer and some might need further explanations. This point could also be valid for other Stack Exchange sites.
Chat is a place to have lengthy discussions that tend to deviate from the original topic. It facilitates a more active discussion than a comment thread (even with live updates) could ever provide.
The system actively encourages this once a post has received a certain number of comments. A message is displayed above the comment box saying:
Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat?
2. To facilitate internal communications
The chat servers are also used internally to facilitate communication between the Stack Exchange team/Community managers and our team of mods. This is a private room that only diamond studded users have access to. It's called The Teachers Lounge.
3. To facilitate private communication with moderators
Private chat rooms can also be used to allow a moderator to reach out to a specific user in order to resolve some issue with their account or some content issue. Mods even have a "super ping" feature that allows them to ping any user in the network even if the user is not currently in chat.