Apple support suggested that I reinstall macOS 10.13.6, which I did following the instructions at https://support.apple.com/en-us/HT204904. That did not fix the problem, and got me escalated to a tier 1 specialist.
The tier 1 specialist said that she had several reports of this today. We verified that the problem occurs in Safe Mode. This got me escalated to a tier 2 specialist.
While speaking with the tier 2 specialist, I remembered that I have an older MacBook Pro running macOS 10.11.6. This same problem occurs on that device. And there have been no changes to that device is months. So I have 2 older (and different) versions of macOS that spontaneously developed this problem.
On my wife's MacBook Pro running Catalina, there is no problem. On my newer MacBook Pro running Big Sur, there is no problem (and no Updates tab in App Store.)
So here's my theory about what is happening. The Updates tab contacts an Apple server, which responds with markup - probably HTML + CSS. I'll bet that markup is not being tested for older version of macOS, and is now broken. If App Store can't understand the markup (probably the CSS), then that would explain why it looks like a bad HTML page. It's doing its best to display HTML that it doesn't understand.
The tier 2 specialist said that there's nothing more he can do for me, and suggested that I post in this forum. It seems like I'll need to either wait for Apple to fix the problem (either on their server or via a patch), or live with an ugly Updates tab.