You can configure automatic emails to be sent to your customers when billing related events happen like an invoice being finalized, a free trial ending, a payment failing and many more. These come out-of-the-box with your Stripe account and below is where to configure those.
Stripe will not send emails in Test-Mode. Enable Live-Mode to fully test automated emails.
If there are notifications you want to send your customers that are not provided out-of-the-box, you can set that up by listening to webhooks. Services like Zapier are an easy and no-code way to listen to Stripe events and perform actions like send an email when the event has occurred.