International Travel: Use of U.S. Domestic Airlines

Travel supported by U.S. Government funds (i.e., federal grants and contracts) must use U.S. domestic airlines for international air transport of people and property to the extent that such service is available.

Essentially what this mean is:
For any leg of international travel originating out of or returning to the United States the federal government requires the traveler to purchase airfare through a U.S. domestic airline (directly or through a travel agent). Note: non-U.S. airlines "operated by" (and ticketed through) U.S. domestic airlines are ok.

i.e., The U.S. Government is stating that international airfares on federally funded grants and contracts be spent supporting the U.S. airline industry (to the extent that such service is available).