To build a contact form in Node.js with SendGrid, developers need to install Node.js and create a free Twilio SendGrid account. They then configure their environment by generating an API key and building the scaffolding for their Node.js application. The application structure includes three files: index.js, form.html, and .env, which securely stores the SendGrid API key. The form is displayed on the main page of the server, and when submitted, it sends a POST request to the /contact route, where the data is processed and sent out as an email using SendGrid. Once the application is set up, developers can test it by entering the URL in their browser and submitting the form, which should result in receiving an email regarding the contact request.