I'm trying to produce the effect in the below image:
I'd preferably like to do this with Photoshop or Illustrator. How can this be accomplished?
If you want a quick and easy solution, I recommend searching for PSD mock-up templates.
I made this in Photoshop using that template:
Here is one that is available for purchase that is a little closer to the one you linked: Psd Business Card Mockup Vol7
From now learning from other PSD mockup maybe the solution. There are a lot of Free PSD Mockup out there like this website get a ton of free mockup http://webgraphic-template.com/31-free-responsive-mockup-psds/