Separation of concerns is a very important tenant to any type of software development, even on the web. Too many times I have found that people just throw everything into as few files as possible and call it a day. This is most certainly the wrong way to do it. As has been mentioned, it will help with maintainability of the code for others, but more than that, it helps you be able to read the code. When everything is separated out, you can think about easily.
Code Ignitor, I have found, has been the easiest to learn framework for working with PHP. I pretty much started my current job and was up and running with it within a few days, from never having heard of it, to using it pretty efficiently. I don't see it as another language at all, either. Basically, using the framework forces me to organize things in a manageable way, and the added functionality is anlagous to using plugins and such for jQuery, or importing packages in Java. The thought that it's like learning another language seems almost silly.
So, in short, organize organize organize. Keep in mind, though, that there is a level of abstraction that just becomes absurd. A rule of thumb is that a class (or file in our case) should do one thing very well. This doesn't mean it is a class that wraps around print, but takes a string, formats it using a complex algorithm and then prints it (this is just an example). Each class should do something specific, and you can do that without any framework. What makes MVC great, though, is that it lets you organize things further, not just on the single class level, but on the level of "packages", being Model, View, and Controller (at least in the case of these frameworks; there are other ways to package projects). So, now you have single classes that do things well, and then you have them grouped with similar classes that do other things well. This way, everything is kept very clean an manageable.
The last level to think about once you have things organized into classes, and then packages, is how these classes get accessed between packages. When using MVC, the access usually will go Model<->Controller<->View, thus separating the model (which is usually database stuff and "business" code in the PHP world), from the view (which usually takes information from the user, and passes it along to the controller, who will then get more information from the model, if necessary, or do something else with the input information). The controller kind of works like the switchboard between the two other packages usually. Again, there are other ways to go with packaging and such, but this is a common way.
I hope that helps.