Subversion is a centralized repository system. That means, whenever you commit to it, you are committing to the hosted repository you have on Assembla.
It seems that you are looking at a locally hosted svn repository if you see "locks, confs, db" as your folder structure. You should be seeing the folder structure that you see in your Assembla Code browser.
You only need to checkout the repository from Assembla to start working, here is a rather outdated video to help you, though it still shows you how to properly connect tortoisesvn to Assembla: http://www.youtube.com/watch?v=IWgTyOOufow