Editing Business card template

I believe I know what your problem is.

Disable the Auto-Select tickbox on the properties bar at the top of your screen.

Basically, each time you're clicking with this on, Ps is selecting the topmost layer.

hth

A mockup template would be populated with your business card front and back business card image files.  You would use Replace Content to replace the content of the Smart Object  Layers object that are in the template PSD if the have the correct Aspect Ratio, Size and Print resolution.   If you card image files doe not have the correct Aspect Ratio, Size and Print resolution the Templates Associated Smart Object layer transform will not work Correctly.  In the Case you can most like open the Smart Object Layer Object and edit its content. Place in or Paste in  you business card front or back. image. And resize for the object size. You need to be careful with the resize if Your cards do not have the same aspect ratio as the templates object.  The  Resulting distortion may not be acceptable.

The Batch Mockup scripts in Mockup Toolkit package can populate your cards files into the template either way Replace Content or Edit Content if the object in the Template are Photoshop object.  If you need to use the Edit option  You cards need to have an aspect ration nearly the same as the one used  by Adobe in their template.    It is also very easy to create your own Template.

Photoshop Photo Collage and Mockup Toolkit

There are layers, however you will need to replace the content and distort it to fit.

What sort of file is it? (i.e. what is the file extension?)

Note: screen shots are useful so thanks for providing it, but it is better to paste your entire workspace so we can see the document information in its tab.

Is it definitely intended for Photoshop?

Where did you obtain the template?  If you provide a link, we can see for sure how it is designed to be used.