How to implement SOLID principles into an existing project
Single Responsibility Principle - A class should have only one reason to change. If you have a monolithic class, then it probably has more than one reason to change. Simply define your one reason to change, and be as granular as reasonable. I would suggest to start "large". Refactor one third of the code out into another class. Once you have that, then start over with your new class. Going straight from one class to 20 is too daunting.
Open/Closed Principle - A class should be open for extension, but closed for change. Where reasonable, mark your members and methods as virtual or abstract. Each item should be relatively small in nature, and give you some base functionality or definition of behavior. However, if you need to change the functionality later, you will be able to add code, rather than change code to introduce new/different functionality.
Liskov Substitution Principle - A class should be substitutable for its base class. The key here, in my opinion, is do to inheritance correctly. If you have a huge case statement, or two pages of if statements that check the derived type of the object, then your violating this principle and need to rethink your approach.
Interface Segregation Principle - In my mind, this principle closely resembles the Single Responsibility principle. It just applies specifically to a high level (or mature) class/interface. One way to use this principle in a large class is to make your class implement an empty interface. Next, change all of the types that use your class to be the type of the interface. This will break your code. However, it will point out exactly how you are consuming your class. If you have three instances that each use their own subset of methods and properties, then you now know that you need three different interfaces. Each interface represents a collective set of functionality, and one reason to change.
Dependency Inversion Principle - The parent / child allegory made me understand this. Think of a parent class. It defines behavior, but isn't concerned with the dirty details. It's dependable. A child class, however, is all about the details, and can't be depended upon because it changes often. You always want to depend upon the parent, responsible classes, and never the other way around. If you have a parent class depending upon a child class, you'll get unexpected behavior when you change something. In my mind, this is the same mindset of SOA. A service contract defines inputs, outputs, and behavior, with no details.
Of course, my opinions and understandings may be incomplete or wrong. I would suggest learning from people who have mastered these principles, like Uncle Bob. A good starting point for me was his book, Agile Principles, Patterns, and Practices in C#. Another good resource was Uncle Bob on Hanselminutes.
Of course, as Joel and Jeff pointed out, these are principles, not rules. They are to be tools to help guide you, not the law of the land.
EDIT:
I just found these SOLID screencasts which look really interesting. Each one is approximately 10-15 minutes long.
There's a classic book by Martin Fowler - Refactoring: Improving the Design of Existing Code.
There he provides a set of design techniques and example of decisions to make your existing codebase more manageable and maintainable (and that what SOLID principals are all about). Even though there are some standard routines in refactoring it is a very custom process and one solution couldn't be applied to all project.
Unit testing is one of the corner pillars for this process to succeed. You do need to cover your existing codebase with enough code coverage so that you'd be sure you don't break stuff while changing it. Actually using modern unit testing framework with mocking support will lead encourage you to better design.
There are tools like ReSharper (my favorite) and CodeRush to assist with tedious code changes. But those are usually trivial mechanical stuff, making design decisions is much more complex process and there's no so much tool support. Using class diagrams and UML helps. That what I would start from, actually. Try to make sense of what is already there and bring some structure to it. Then from there you can make decisions about decomposition and relations between different components and change your code accordingly.
Hope this helps and happy refactoring!
It will be a time consuming process. You need to read the code and identify parts that do not meet the SOLID principles and refactor into new classes. Using a VS add-in like Resharper (http://www.jetbrains.com) will assist with the refactoring process.
Ideally you will have good coverage of automated unit tests so that you can ensure your changes do not introduce problems with the code.
More Information
In the main API class, you need to identify methods that relate to each other and create a class that more specifically represents what actions the method performs.
e.g.
Let's say I had an Address class with separate variables containing street number, name, etc. This class is responsible for inserting, updating, deleting, etc. If I also needed to format an address a specific way for a postal address, I could have a method called GetFormattedPostalAddress() that returned the formatted address.
Alternatively, I could refactor this method into a class called AddressFormatter that takes an Address in it constructor and has a Get property called PostalAddress that returns the formatted address.
The idea is to separate different responsibilities into separate classes.