Double click a task to open task edit dialog. Start typing into Resource field and it will provide a list of matching users. Select a user and click Save button.
In order to make a task unassigned click the “x” on the right side of Assignee input field to clear it, and click Save.
Depending on chart settings, the Resource field can read/write data from Jira Assignee field, or from another Jira custom User picker field.