To do this you need to play as Snake while playing on his stage, Shadow Moses Island. If you have your taunts set to the d-pad on either the Gamecube controller or the Wii controller, press down on the d-pad repeatedly. This must be done extremely fast.
If executed properly, Snake will take a knee and get on his Codec, where he'll speak with familiar characters from the Metal Gear Solid series, including Otacon, Colonel Campbell, and the like. You'll have a completely unique, often humorous conversation with the person on the other end of the Codec depending on who you're fighting. That's right -- there are completely unique conversations for each character, including character variants (Shiek, Zero Suit Samus, the various Pokmon). Give it a try!
Additionally, if Snake is killed when using the Codec, another interesting sequence occurs.