NOTE:
This only applies to campaigns created under the new system. It's up to the campaign designer to make sure all the files have been structured and organized correctly.
See this thread for details on configuration:
viewtopic.php?t=29080
Also: Don't worry about my desktop image in the background there.

Special thanks to nikivdd for letting me use his custom campaign as a template.
Step 1: Download someone’s campaign.

Step 2: If you don’t have a ‘Mods’ folder, create one.

Step 3: Place downloaded file into ‘Mods’ folder.

Step 4: Start Panzer Corps and press this button.

Step 5: Enjoy!

Now, tell me that wasn't painless and easy.

We strongly encourage our custom campaign designers to make the most of the new system. One of the best features over the old system is the ability for custom campaigns to be installed in parallel, no more constantly overwriting campaign.pzdat files or other files otherwise only contained in the DATA folder.