You are using the wrong sense of the word 'free'. Do you speak French at all? 'free' in open source terms is 'libre' in French, not 'gratuit'. In English the usual comparison is 'free as in speech. not as in beer'.

Android is free in that anyone can download and compile their own version of Android and can distribute that version as they wish. If you use one of those distributions of Android you can have root access on the phone if you wish. Manufacturers are free (i.e. at liberty) to install Android on their phones whether or not they pay Google anything.

However, although the operating system is free, some of the applications are not. If you distribute your own version of Android you have to omit or substitute some of Google's applications. One of the non-Google distributors of Android (cyanogen) recently ran into trouble because he was including Google's closed-source applications in his distribution. The most obvious missing link if you don't use a Google distribution is the marketplace app: you can still install software without it, but you cannot install paid applications.

That may not be a problem if you are a phone manufacturer: just have your own version of the marketplace which only allows access to your own little walled garden of apps. That's probably quite appealing for some phone providers.

As to how they can do this:

All Android applications must be digitally signed. This allows other applications or servers to identify the author or the application (or rather the person who compiled it). Even if you have the source for Google's marketplace the servers will not identify a copy you compile as being a valid Google copy of the application, so it only gets to see the free apps.