Unfortunately, I don't have a tutorial here for doing this, and it doesn't look like there is an easy built-in way to do this. The way I see it, though, you have a couple of choices. First, you can embed your game in a C# Windows Form application. Then you can use all of the normal windows forms stuff, including a file browser. I've done this before for the Evolution Project, and it can be kind of a lot of work to do, but it also gives you a lot of power. If you want to try this, you might want to check out this link on Nuclex: [http://www.nuclex.org/downloads/developers/kits/xna-usercontrol]. I kind of don't think this will work on an Xbox though.
Your other choice might be to create your own file browser from scratch using the File class, the Directory class, and maybe some other stuff in the System.IO namespace to navigate the file system, and do all of your own rendering. This method, too, will be a lot of work, because you would basically have to create your own file dialog class.
On a completely separate note, I wish that someone would sit down and make a library of components that are equivalent to windows forms, but are completely XNA based. This would include buttons, progress bars, scroll panes, and yes, even file dialogs — all of which would work in an XNA only environment.
Hopefully this helps, or at least gets you going in the right direction.