An MVC framework is a framework written to split up the the business logic, database access and presentation.
This is very useful in most web applications, and now lately into software/desktop applications.
This is due to the fact that following the MVC model, your code will be much clearer, cleaner and you keep your application DRY (Do not Repeat Yourself).
You can write your own classes and separate them into Model, View and Control. But again, you will need a framework to help you in accomplishing certain tasks. Like a List control in ASP.NET, or PHP framework being able to help you translate text between languages and so on. (Oh why reinvent the wheel?!)