The Open-Closed Principle is a design principle that states entities should be open for extension but closed for modification. In the context of Laravel and Twilio SMS, this principle helps to write clean, reusable, and maintainable code by separating extensible behavior behind an interface and flipping dependencies. The solution involves creating an SmsInterface that defines the sendSms method, implementing it in a TwilioSms class, configuring and registering the Twilio service, and modifying the SmsController to use the interface instead of the concrete implementation. This approach allows for easy extension with other SMS providers without modifying the controller class.