What does an incognito browser have to do with anything?

An incognito browser would ignore all client-side cookies, so the Slack web client would not try to - say - resume a previous user's session or re-use any previously saved data.

Likewise, incognito mode will also ignore most cached web content, meaning all assets on the Slack web app will get loaded again from scratch. This "clean state" start could, theoretically, get around issues with old - potentially incorrect/outdated - assets being loaded, even though that really shouldn't happen under most circumstances.

Sure, but why does that indicate the issue probably wasn’t related to a code push, like the person I responded to said?

It means that one is not sending a session cookie of any kind, thus should be sent to a 100% cached version. No "Are you XYZ and what to log into ABC's Slack again?" box.

That means it's not a user auth error

non-logged in user may not go through all the same codepaths as a user with cookies present.