If, after completing a task, you want to remove it from your project board, the best way to do that is by archiving the task. This way, the task will be moved to the project's task archive, where it can still be reviewed and, if necessary, made active again anytime.
There are two ways to archive a task:
Archive After Completion
Active tasks have a big Complete button in the upper left corner of the task dialog.
Once you click on the Complete button, it turns into an Archive button, which you can click to archive your completed task.
Archive Without Completion
If you want to archive a task without completing it first, you can click on the ellipsis in the upper right corner of the task dialog, and simply select Archive from the drop-down menu.
By the way, you can find more information on task statuses available in MeisterTask here.
Reviewing and Retrieving Archived Tasks
To review or retrieve archived tasks, click on Active in the top bar of your project board. This will open a view switcher, where you can jump back and forth between your active, archived and trashed tasks. Select Archived.
In the project archive, you'll see a list of all your archived tasks, and the filter sidebar to help you find previously archived tasks. Click on any task to view its details.
To move an archived task back into your active project, click on the ellipsis in the task dialog and select Restore Task from the menu.