In short, with the stationary camera, a still image of the set is captured. This still image is then split into halves* (one half for Andy's chair, and one half for Conan's chair).
The two "empty desk set" photos are overlaid onto the live camera image. For the start of the bit, the opacity is set to 0% (transparency at 100%).
During the bit, the half with the Conan "empty desk" picture slowly has its opacity increased (reducing the transparency) - thus providing the feeling that Conan is disappearing. (The same process is applied for Andy later in the bit). As we see more of the "empty desk" photo, we see less of "live Conan".
Both actors are in their normal position on the set during the bit. The audience, in order to understand the joke, must look at the monitors inside the studio.
*Note: It may not truly be a "half", based on the needs for special lighting used later in the joke. In fact, if you look at Conan's finger tips on the desk - you can see a bit of the "overlay cutout" leaving his fingertips at full opacity for most of the bit. Though, it looks that way towards the end when Conan has his overhead lighting increased and Andy is fading out.