Dropbox has an API that allows developers to download and store files from publicly available URLs, making it useful for integrating fax services like Twilio. To send a fax to Dropbox, users need a fax-enabled number from Twilio, a Dropbox account, and set up a new app on the Dropbox developer platform. The process involves creating a function to receive faxes and uploading the received file to Dropbox using the API, as well as configuring a TwiML bin to handle incoming faxes and sending a test fax using the Twilio API or a library's explorer tool. Once completed, users can check their Dropbox folder for the uploaded file and use this integration to build various fax-related applications.