Many pictures on this website were made the free software blender. This page contains some hints about how to use the corresponding files to get a better visualization than in still pictures. Its purpose is not to explain how to produce them. If you do not have a pretty recent version of blender, you first need to get it on blender's download page. Then you can get an example blender contact geometry file and a description of its content on my convex surfaces page.
General caveat: The blender interface can be divided in arbitrarily many editors (sub-windows). It is important to understand that all hotkeys are relative to a particular type of editor so being careful about the position of the mouse cursor when pressing keys saves a lot of frustration.
The following keys allows to move the viewpoint:
The basic way to select an object is to Right-click on it. See the select menu for a lot of other ways. An important type of editor is the outliner. It shows a list of objects in the current scene. Clicking on the eye icon allows to hide or show an object. The only surface of the xit3.blend file which is not hidden in the beginning is the example of foliation realization, try using the outliner to see the others.
The file elimination.blend explains Giroux's elimination move. It contains an animation which is easier to understand by moving in time by hand, changing the view point as needed. Here we have a new editor type, the timeline: Clicking and draging in this zone changes the displayed frame. Frame 1 shows a surface having a pair of singularities in elimination position which has disappeared on frame 200.