Make sure the right split is "active" by clicking in its main content area. Then select a view from the tree on left, which will create a new tab for the selected tree item in the right (active) view.
To demonstrate both sides playing simultaneously, I have my music library playing on the left, and the TV Show Dead Ringers on the right. You can tell which view is active/inactive by the color of the tab bar (the inactive tab bar is typically a "washed out" background color, but that appearance varies and depends on the particular skin being used).
In the attached picture, you can see I have a bunch of podcast tabs (which are also locked) - the TV show is in the "new" and unlocked tab (my lock symbol is just a dot). Podcasts is what I personally use for the right-hand split.
Hope that helps.