Prerequisites

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

Guide

1. Go to Automation listing page (it is either Project settings > Project automation or Jira Settings > System > Automation rules) and press Create rule button:
   
   
   
   
2. On the New trigger page select Field value changed option:
   
   
   
   
3. On the Field value changed page select Checklist Completed (Field) in the first list and next select Edit issue in the second list, and press Save button:
   
   
   
   
4. On the Add condition page select New condition option:
   
   
   
   
5. On the New condition page select Issue fields condition option:
   
   
   
   

6. On the Issue fields condition page paste select "Checklist Completed" from the list of fields, "equals" in the condition dropdown, and "Not Completed" in the value dropdown. Next, press Save button:
   
   
   
   Hint: Please note that adding a new item to the already completed checklist, will make the checklist uncompleted. 
   
   

7. On the New component page select New action option:
   
   
   
   
8. On the New action page select Transition issue option:
   
   
   
   
9. On the Transition issue page select desired transition status, e.g. In Progress or any other status that you want, and press Save button:
   
   
   
   
10. On the next page, add a name/description to the automation rule and press Turn it on button:
    
    
    
    
11. Test your setup and enjoy: 
    1. go to any issue page,
    2. add a checklist,
    3. complete the checklist (check all items)
    4. transition the issue to some other status than selected in this tutorial
    5. uncomplete checklist (open one item) and wait a moment (Automation needs time to process the data)
    6. Next, refresh the page and notice that issue is in the desired status (e.g. In Progress).