To send email attachments with Twilio SendGrid and PHP, a developer must first install the necessary packages such as PHP version 7.0 or higher, Composer, and the `sendgrid/sendgrid` package to access the SendGrid API. A free SendGrid account should also be created, followed by obtaining an API key for authentication purposes. The `vlucas/phpdotenv` package is used to load environment variables into the application. Once the API key is obtained, it can be stored in a `.env` file instead of directly in the code for security reasons. The developer then uses PHP to create a SendGrid Mail object and adds attachments to the email using base64 encoding. Finally, the email is sent to a recipient by calling the `send` method on the SendGrid API instance, and the response status code is printed to the console to verify successful transmission.