To block spam calls with PHP and Laravel, developers can use Twilio's Add-ons Marketplace to integrate third-party technologies that filter out unwanted callers. The process involves installing an add-on such as Marchex Clean Call or Nomorobo Spam Score, configuring the add-on to work with Twilio's API, and writing custom code in PHP to check the spam score of incoming calls and reject spammers accordingly. The code uses Laravel middleware to handle incoming requests and TwiML responses to control the flow of calls. Once set up, the application can be configured to call a public host or use ngrok for local testing, and developers can test blocked calls using unit tests with dummied-up JSON data.