To checkout the subversion repository you have to right click the folder you want to download the files to in your explorer window, and select the Tortoise->Checkout menu.
Just enter https://ufoai.svn.sourceforge.net/svnroot/ufoai/ufoai/trunk in the URL of repository as shown in the image below. Accept the certificate.
To update your version of the source to the latest revision, right click the folder and pick SVN Update.
To upload new files within the Template:Path directory you should just mark the file and click it with your right mouse button, choose Tortoise->Add and then Tortoise->Commit to check it in. But please also note the Coding guidelines about svn commit logs.
Caching hogs resources
You may find disk IO grind to a halt. If you see TSVNCache.exe is the culprit in task manager. take some advice from this thread. Disable-TSVNCache.exe-to-minimize-disk-IO