The theme you are using is probably signing users up to the Shopify customer list.
There's no built-in way within Shopify to welcome new email subscribers, but you can use MailChimp to do this. You'll need to use an app called ShopSync to sync your Shopify contacts and data with MailChimp, so that you can create the automated email in MailChimp.
How to link Shopify with MailChimp:
How to setup a welcome email in MailChimp:
This should get you up and running!
To set up an automated welcome email for people who can subscribe to your newsletter you have to install ann app in order to send automated welcome emails.
You can install AiTrillion.com. It has powerful customer segmentation, custom drag and drop email editor, AI-enabled email scheduling, which can help you to send targeted emails, other than that it also has workflow automation. You can use this platform to send email based on their buying behavior along with all these features, it also has prebuilt workflow and templates.
It has integration of email marketing with loyalty reward program. You can give loyalty points to your customers on signing up for the newsletter which will help you to increase the number of newsletter subscribers. Email marketing is also integrated with reviews, reports and analytics, smart pop-ups. You can check it out.