1. Install TortoiseSVN from their downloads page
2. Make a folder on your hard drive where you want to check out the sources. You can call it BetterAI or BetterAITrunk or BetterAIVanilla or whatever you want. (If you like, you can make a folder called BetterAI and then a folder inside that called warlords and a separate folder called vanilla). (Another way to do it is to check out the whole project, but this will be a lot of versions, it is not recommended unless you have a lot of disk space).
3. After making the folder(s), however you want to organize them, right click the folder into which you want to put the sources, and select "SVN Checkout..." from the menu. (You will have this choice if you installed TortoiseSVN properly).
4. In the window that comes up, for "URL of repository" you want one of the following:
1. "https://svn.sourceforge.net/svnroot/civ4betterai/trunk/": to check out the latest version to run on the warlords codebase
2. "https://svn.sourceforge.net/svnroot/civ4betterai/branches/vanilla/": to check out the vanilla version (that is the 1.61+ version), this code may be slightly behind the trunk, as it has to be manually syncronized. It is generally about a week behind.
3. "https://svn.sourceforge.net/svnroot/civ4betterai/": to check out the entire project, this will be over 6 complete versions of the code (one for each branch or tag), it is not recommended.
5. Click "OK" and the sources will begin downloading!
6. To update to the latest sources, right click the folder again and just select "SVN Update" and it will automatically update you. You can click on the "Show Log" button (or select that from the menu) to read the individual checkin comments and/or get a list of which files were changed with each revision.