The technique goes by different names: freeform, Manhattan, dead bug, air-wired. Whatever you call it, it involves building your circuit by bending wires to form traces without using a circuit board.
Similar techniques pre-date PCBs. If you ever took apart an old TV or radio, you have probably seen this type of construction. This modern version is elegant and beautiful (for a given geeky definition of beautiful). It is especially impressive that it even sometimes includes surface-mount components.
Here is a great example project of a see-through Arduino pictured above. Check out the full write-up on Instructables.