With Twilio and Java, users can make a mysterious phone call that plays a chosen sound file to any phone number without having to pick up their mobile device. The process involves creating a directory for the project, downloading the Twilio Java helper library, buying a Twilio phone number, hosting an MP3 file publicly using the Twilio Assets dashboard, and configuring environment variables with account credentials. Users then create a Java class that initializes the Twilio client, creates a new call instance to make an outbound call to their personal phone number from their Twilio phone number, and deploys the audio file publicly to be played during the call. The code can be compiled and run using various commands depending on the operating system, resulting in a mysterious phone call that plays the uploaded audio file when answered.