To manage Go application secrets using Vault, developers need to use a tool like HashiCorp's official Go library. First, they need to set up Vault by running `vault server -dev` and then setting environment variables for the address and token. Then, they can create a new project directory and install the required dependency. Next, they can write code that uses Vault's Key/Value storage engine to store and retrieve secrets. The code provides examples of how to store a secret, retrieve a secret, retrieve all versions of a secret, delete a single version of a secret, and delete all versions of a secret. It also notes the importance of security when using Vault and provides guidance on how to use it in production applications.