not sure about the fat one, but for the slim model you have to first get swap magic + magic keys, then use the said keys (which are just few pieces of specially modeled plastic) to cover the sensors in the ps2 which detect whether the lid is open or not. there are 3 of those.
once covered boot the ps2 using swap magic dvd without closing the lid (with the magic keys the sensors won't notice the lid is open and will allow you to run stuff with open lid, thus allowing to swap the dvd later), then once the booting is done and you're in swap magic's menu the dvd will stop spinning. swap it now with your game, then press start game in the menu to play. you can now close the lid.