To generate Twilio Access Tokens in Swift, developers can use the Vapor web framework and a small helper library to return tokens from a server. First, set up the project with Vapor, Swift Package Manager, and the necessary dependencies, including the TwilioAccessToken and Environment packages. Then, create a new route for generating tokens that takes an identity and device as parameters, verifies the environment variables, creates a token, adds a grant for Programmable Video if desired, and returns the token in an HTTP response. Finally, test the server by making a GET request to the /token route with the required parameters and verify the decoded token using a JWT Debugger.