An Action is a "container’ of sorts, a datablock that contains keyframe data, first created when anything is keyframed on an object. As an Action, it can be linked in the Action Editor to any other object. Deleting its contents, the keyframes, does not automatically delete the Action, just as emptying a jar doesn’t destroy the jar itself.
First, select the object that it’s linked to (“vaeg,” seen in the Outliner image), then Shift+LMB (click) on the “X” button next to the Action name in the Action Editor. This will unlink the Action from that object and also mark it for deletion when the file is saved, closed, and re-opened. If the Shift button is not included, it will unlink the Action from the object but the Action will not be deleted.
This may seem a bit arcane but there are good reasons for Actions to have persistence both when unlinked and when empty of keyframe data – in the latter case, they usually get filled back up ;).