Why should I use this software instead of, say, Microsoft Project? What kind of level of familarisation with TFS does it demand? Is there anything similar I could use with SVN instead?
I think eScrum has gone long time ago. In TFS 2008 Scrum for Team system totally dominated and in TFS 2010 MS published their own template. Comparing it with MS Project doesn't make much sense because these templates are about setting methodology in a project development.
These templates consists of prepared TFS artifacts which allow you creating and controling product backlog and items (user stories), spring backlog and items (tasks), sprints, releases, etc. It has also built-in reports like product burndown and sprint burndown charts.
Integration into TFS allows developers associating their code commits with particular tasks and completion of tasks automatically track completion of related user stories. It also allows tracking bugs and new backlog items related to already completed user stories. TFS artifacts are also customizable.
If you use TFS it is probably good choice but if you don't you should check another product because using these templates make sence only if you migrate whole development process to TFS.
I tested OneTime 2010 with Visual studio but I never had chance to use it in the real project. They also have some integration with SVN by using their own SVN plugin to Visual studio. There will be a lot of other products available.