It’s so obvious to everyone else, that I couldn’t find any reference to it on the internet: you have to solder an Arduino shield to the headers.
I thought it was plug and play, since the little header pins seemed very snug when I put the shield over the Arduino Uno. But the shield wasn’t working, and it turns out that each pin needs to be soldered.
Luckily, soldering is very easy, and I was able to do it in less than 10 minutes, never having soldered before. It’s worth checking out some YouTube vids before you start, though.