Using SVN for
Download and install Tortoise SVN if you don't already have it (http://tortoisesvn.net/downloads.html)
Prepare a folder to contain your copy of the project, it will be known as your 'Working Copy.' This Working Copy will be the local directory on your machine (that you just created) which will be your 'image' of the central repository - this is where you will make your changes when you want to contribute them and where other people's changes will sync down to). I strongly recommend that this is NOT your 'live' copy in 'program files/firaxis/...'. Mine is in \development\RoMAND, but the location doesn't matter - just put it somewhere convenient where you have space.
Checkout from the repository to the folder you created in (2). This will get the current revision and make a permanent connection between your working folder and the repository. Assuming you have installed SVN as in (1) you can do this by right-clicking the directory in Windows Explorer and selecting 'Tortoise SVN/Checkout...'
When it asks you for the repository URL enter:
'https://anewdawn.svn.sourceforge.net/svnroot/anewdawn/Trunk/Rise of Mankind - A New Dawn '
: Don't highlight and copy paste this, instead ~ right click and copy link address.
Future refreshes or uploads (aka Commits) only get changed files, so they will be much, much faster, especially since the really big things like art assets and movies don't tend to change much.
After it's all copied you are up and running!
Now how to get the working copy into play?:
1) Create an empty folder somewhere not inside 'program files'
2) Right click and drag the root folder of your working copy to this new folder when you let go of the right mouse button you can..
3) Select the 'export versioned items here' option
4) Move the result into your mods folder (and rename it to Rise of Mankind - A New Dawn if needed)