There are three things that Hide Owner will adjust.
1) At the very top of this image, the URL is circled. Hide Owner will remove the nickname from your URL and make it completely SmugMug.
2) You can see that the custom Banner has been replaced with the SmugMug banner. This will also remove all customization on the page.
3) There are no links to the homepage any longer, just the gallery name.
Click on the photo to see these things more clearly.
If you go into the gallery customization page and change Hide Owner to No, it will change these things back.
Here is the same gallery, with Hide Owner turned off.
Click on the photo to see these things more clearly