When a chat request comes in, it is either authenticated or unauthenticated. If the chat request is authenticated, then Live Assist knows exactly which contact record in Dynamics belongs to this chat customer.
You can click on the visitor name at the top of the Live Assist widget to open the chat context, then click on Details to open that chat in the CRM..
As of Live Assist for Microsoft Dynamics 365 version 3.6.0 (April 2018 Release), Contact information can be loaded by clicking the visitor's name in the Agent Application, as in the following example:
Handling Unauthenticated Visitors
If the chat request is unauthenticated, then Live Assist cannot know exactly which contact record to associate, but it can collect some identifying information (for example, first name, last name, email) using the pre-chat survey. In this case, the visitor's name will appear at the top of the Live Assist widget, but clicking on it will not bring up a contact record. Instead, this will launch a contact search on the first and last names specified by the customer.
The agent can then click on a search result, and the contact record will appear in place of the search.
To associate the contact that you have found to the current chat activity, click on Associate to Chat.
See also: Agent UX Associate CRM Contact with a Chat