After reading some of the answers, I would like to say first that you don't often have much of an option on what framework your going to use or whether your going to use one or not (as a professional). I don't want to intimidate you, but you will need to build enough of an understanding of PHP to allow moving from one framework to another as well using more than one framework at once.
when do I need to use a PHP framework such as CakePHP?
There isn't a universal answer to this. As a professional, your going to be tasked to solve specific problems and this will be the deciding factor for the whens' and whats' of each project, especially if a client has a specific demand. But generally speaking, you will always be using some form of framework with PHP, unless your modifying or extending a pre-existing application that has it's own framework (such as Wordpress or PHPBB).
What are things that this and other similar PHP frameworks offer for me?
As said by dirk, they offer abstraction. What this means is that they take common, multi-step procedures and try to make them as simple and streamlined as possible. In short, they try to present just-add-water solutions for common tasks.
And is it really important to use a framework to be a professional?
Generally speaking, no it is not important to use a framework to be a professional. But realistically, you won't have much of a career if your not familiar with at least a couple common frameworks. PHP without a framework is like C++ without a std library. You don't need it, and you can actually do more without it since you'll be directly using the lower-level builing-block functionality of the language, but it will take you much longer to do the same thing.
And can I create my own framework to provide the features I like into it?
Yes. In fact, you will likely end up doing so even if unintentionally. By this I don't mean an expansive do-all framework, but more like a tool-bag of basic functions and scripts that do tasks aren't common or generic enough to be handled by the average framework. Mine is basically made up of various file system, process handling, and rex-ex tid-bits with a few different curl and stream/socket utilities.