All the Juce source code is hosted on SourceForge - the main page for the project is here.
The latest downloads are available here, where you can get a zip file containing all the source code and examples.
As well as the released versions, you can use the GIT to download the working tip of the source tree. The tip is normally pretty stable, and is worth trying if you're looking for a bugfix that's been done since the last proper release.
To grab the latest version with GIT, just use the command:
git clone git://juce.git.sourceforge.net/gitroot/juce/juce
To browse the repository online at SourceForge, click here.
To install the code, just unzip or it into a convenient location.
There are project files for XCode, Visual Studio, etc. in the juce/build directory,
and a good place to start is the demo application, found in juce/extras/juce demo.
To compile an application that links to the library, you can either look at the projects used by the
demo apps, or there are some basic 'hello world' projects in the juce/extras directory.
More detailed documentation about setting up the build environment for specific compilers is included
in the juce/docs directory of the source-tree.
A complete doxygen-generated API reference guide is available for browsing here.
A zip-file containing the complete API guide is included in the juce/docs directory when you get the tree using GIT, or you can download it directly from here.
The Juce source tree includes a simple application (in juce/extras/juce demo) that shows off a few of
its features. Some pre-compiled binaries of this demo app are available here:
The demo is a good place to look for examples of how to use the basic library components.
The Jucer is a UI component design tool that produces Juce c++ code. The Jucer source code lives inside
the main Juce tree (under juce/extras/the jucer). You can compile it yourself to run on Windows, Mac
and Linux, or download a pre-compiled binary to experiment with:

As well as being a useful tool, it demonstrates many of Juce's features, including things that aren't in the juce demo, such as undo/redo, keyboard shortcut mappings, multi-document interfaces, etc.