I must be missing something here. Since when does earning income on people who sign up on your link at the end of your article have anything to do with asking family and friends to join Medium.

All you have to do is click a box that says something like "You can sign up here for Medium" at the end of your article. That's it.

Medium is highly ranked on Google, so all our articles are sent traffic by google. As they get the first three articles free, they can read your story. After that, they have to pay a subscription.

You're not doing anything to get subscriptions. You've just joined the program, i.e. ticked a box, to have that link under all your articles. That's it.

How are you referrring family and friends? It's complete strangers that are signing up using your link.