Where do alpha testers download Google Play Android apps?

You need to publish the app before it becomes available for testing.

if you publish the app and the apk is only in "alpha testing" section then it is NOT available to general public, only for activated testers in the alpha section.

EDIT: One additional note: "normal" users will not find your app on Google Play, but also the activated tester can not find the application by using the search box.

Only the direct link to the application package will work. (only for the activated testers).

Here is a check list for you:

1) Is your app published? (Production APK is not required for publishing)

2) Did your alpha/beta testers "Accept invitation" to Google+ community or Google group?

3) Are your alpha/beta testers logged in their Google+ account?

4) Are your alpha/beta testers using your link from Google Play developer console? It has format like this: https://play.google.com/apps/testing/com.yourdomain.package

1. Publish your alpha apk by pressing the submit button.

2. Wait until it's published.
   (e.g.: CURRENT APK published on Apr 28, 2015, 2:20:13AM)

3. Select Alpha testers - click Manage list of testers.

4. Share the link with your testers (by email).
   (e.g.: https://play.google.com/apps/testing/uk.co.xxxxx.xxxxx)

It should be noted that releasing an alpha app for the first time may take up to a few hours before an opt-in link is available and invitations are sent out to the email addresses in your testers list.

From Google support:

After publishing an alpha/beta app for the first time, it may take a few hours for your test link to be available to testers. If you publish additional changes, they may take several hours to be available for testers. [source]

You may want to wait until you have an initial opt-in link before publishing more changes to the app because doing so is likely to increase your wait time for receiving your tester link; or, may lead to your testers testing with the incorrect version.

Hope that clears things up for anyone confused about why they don't have an opt-in link as depicted in screenshots in this SO thread!