You are using Email to Case Premium and need to set up mail routing into Salesforce from your G-Suite account.

By configuring your G-Suite account to forward messages to Salesforce, you can allow your customers to create and update cases by sending email to an address within your email domain.

For those using G-Suite, you may come across a number of different ways to forward messages from one address to another. When setting up forwarding between G-Suite and Email to Case Premium, Internet Creations recommends using Routing.