The answer is:
1. Go to Apple's App Store app on your iOS device.
2. Search for your magazine name.
3. Click the "Open" button.
4. Find the "In-App Restore" button inside the app and click it.
5. Download the current issue of your magazine.
The reason for the confusion is that many magazines had two things on an iOS device: a magazine subscription housed in the Newsstand app and a separate standalone app branded by the magazine.
Every time I read about downloading the app, I thought that the writer didn't understand my question. I was looking for the latest issue of the magazine which I purchased through Newsstand, not the magazine's app.