Hi.
Your website must be online and accessible in order for it to be able to receive webhooks from Stripe. Also webhooks must be properly configured as well. Last, but not least – your plan must be properly configured to be “$20 monthly for 3 months”.
What values are you using for your subscription plan so we can confirm that the correct parameters are being used to make it collect only 3 payments then stop?
If all the above is correct, then debug log should have some insights on what’s wrong. Also you can check subscriptions status by going to WP Dashboard -> Stripe Payments -> Subscriptions. Click any and see if there is anything useful available on events log.