This guide is based on Jira built-in automation functionality.

The guide works for Classic projects only. Next-gen projects are too limited and do not support automation features.

Prerequisites

"Save checklist data to Jira custom fields" option must be enabled in Issue Checklist Global Settings.

Step by step guide

Create a new Automation rule (go to Project Settings > Automation and click Create rule button)
Select "Issue Transitioned" as a trigger and next choose source and destination statuses (details of transition)
Select Action as the next step and choose "Edit Issue" from the list of actions
Select Checklist Text field on the next screen
Paste checklist in text format as presented on the screenshot, i.e.:
Save the rule and test your setup (transition an issue, refresh the page and spot the checklist updated)