FAA JO 7210.3X Change 1 defines it in 17-14-2.http://www.faa.gov/regulations_policies ... ID/1019804
SWAP statements are prepared by ARTCC (center) Traffic Management Units as deemed necessary anytime there is adverse weather that would affect one or more routes of flight through their coverage area. They can include any combination of Coded Departure Routes, playbook, FEA/FCA, tunneling, ground stop, ground delay programs they would like to be considered. These are sent to the ATCSCC (system command center...where kyle works) to be discussed and coordinated, to be sent out as a SWAP advisory. This advisory goes to all facilities that are affected by said traffic management initiatives.
What you are probably familiar with is the "National Playbook" which has predefined reroutings that are commonly used. These are a collection of SWAP routes that have been validated by ATCSCC and make for easy coordination with impacted ARTCCs. However, SWAP can include many items outside the playbook.
Simple answer to your question: Center TMUs send request to ATCSCC which distributes to affected facilities. The managers at said facilities tell the controllers to solicit pilot compliance with request.