To get started, versions must already exist in your project. 

Click on Version Planner at the sidebar of your Jira project.
The regular board view will appear on the right. But instead of displaying the workflow status, the columns show the versions available in the project.

If you've set a release date for a version, it will also be displayed within the version column.

All upcoming releases are now displayed and additionally a column that shows issues that are not assigned to any release version yet.

Use the dropdown in the upper left corner to filter the board view based on the release status and / or issue types.

And here is the actual benefit of the Version Planner.
To plan a release, click on an issue and drag it via drag'n'drop to the matching version column.

This will automatically update the field value for "fixVersion" within the issue. 

If an issue has multiple tagged versions, it will be displayed in multiple columns on the Version Planner Board. By changing one of those, the other tagged versions will still remain. 

That's it! Happy Release Planning!