That functionality is disabled by default. It can be enabled by switching off the "Save checklist data to Jira custom fields" toggle in Global Settings page.
When "Save checklist data to Jira custom fields" option is enabled, then checklist changes are recorded in the user's Notifications area as presented on the screenshot:
Changes are tracked as the Issue Checklist user. If you would like to see a real user in the issue history, please visit the Checklist History tab.