Options to download Garage Band app for macOS unsupported in App Store?

There are currently two entirely different GarageBand apps in the macOS App Store:

GarageBand 6.0.5 aka GarageBand '11
https://apps.apple.com/us/app/garageband-6-0-5/id408980954?mt=12
This was the GarageBand app released with iLife '11 package back in 2010, in the 6.0.1-6.0.5 version range, and is probably what a lot of long-time Mac users have already. It is a 32-bit app and so won't work on 64-bit-only operating systems like macOS Catalina, Big Sur and Monterey.
This app has the Package ID: com.apple.garageband
GarageBand
https://apps.apple.com/us/app/garageband/id682658836?mt=12
This is the modern GarageBand app that was released with OS X 10.9 Mavericks back in 2013, currently in the 10.0.0-10.4.4 version range.
This app has the Package ID: com.apple.garageband10

Because it has a different Package ID, GarageBand 10.x is not considered to be an upgrade from GarageBand 6.0.5 by the Mac App Store, but an entirely different app. If you never "purchased" the free GarageBand 10 app before Big Sur came out you're not going to be able to click the Get/Install buttons for it on Sierra-Catalina now due to the store validation logic throwing the error message:

We could not complete your purchase.

GarageBand can't be installed on "Macintosh SSD" because macOS version 11 or later is required.

To get around this you need to get some screen time on a Big Sur or Monterey system:

Open the Mac App Store there.
Sign In using your Apple ID.
Get/Install the GarageBand (10) app. You don't need to let the download finish, you can cancel it.
Sign Out of the App Store.

Back on your own Mac App Store (on Sierra through Catalina) you should be able to go to your App Purchase History (click on your Apple ID in the bottom-left corner of the window) to see the new GarageBand app at the top of the list, being your most recent purchase. You may need to Sign Out/Sign In a couple of times, or even reboot your Mac, but it should appear within a few minutes.

Now, there are two ways you cannot install GarageBand 10:

You cannot click on the download button from the app Search results.
You cannot click on the download button from the GarageBand app page.

If you try either of the above methods you'll be greeted with the same error message as earlier:

We could not complete your purchase.

GarageBand can't be installed on "Macintosh SSD" because macOS version 11 or later is required.

There's only one way to install it:

From the App Purchase History page click the download button.

You'll be greeted with a different dialog window:

Download an older version of GarageBand?

The current version requires macOS 11.0 or later, but you can download the last compatible version

Click the download button. On macOS Catalina when the download completes you'll have GarageBand 10.3.5, which is the last compatible version. GarageBand 10.4 and later require macOS Big Sur.

I prefer the last option of another answer.