First off, you need to have the Jetpack plugin installed and have it connected to a wordpress.com account. Jetpack plugin is made by the same folks that run wordpress.com.
Activate Jetpack plugin and connect to your wordpress.com account and then go to Settings. Navigate to your WordPress Dashboard Go to Jetpack > Settings > Discussion.
Scroll down to Subscriptions and turn on.
Go to Appearance > Widgets
Drag over a Blog Subscriptions (Jetpack) Widget to desired Widget area in a location you choose. Click Save. Click Done. See the animation below to see it in action.
You’re done! Go to your website and scroll to the widget area and you should be able to see the subscription box.