It is thought that the mountain lion whistle mostly occurs during twilight and is a form of advertisement or it signifies frustration. By “advertisement” I presume this to mean a call to tell others of the cat’s presence.
The mountain lion (puma) whistle is the most frequently heard sound made by the mountain lion. Stanley Brock (I not sure of his profession) kept a tame female puma called Leemo. He said this about the cat’s use of the whistle:
“Every time I approached her, or passed close by, I would call her by name, and very often give her a tidbit as well. She invariably responded with a high pitched, cheeping, whistle. It is a very difficult sound to imitate but if you spell the meow of a cat with a wh instead of a m and try to whistle the sound instead of speaking it….Later when Leemo grew up, the call became loud enough to be audible on a still day at 300 yards.”
In the San Andres Mountains of New Mexico people studying the puma heard high-pitched whistles from a group of pumas from 300 meters. An orphaned puma also whistled.
This information is a little vague even though it comes from a great source1 dated 2002. It would seem that scientists at that time were unsure as to why mountain lions whistle. The internet at 2016 does not really help either.
Associated: a page on all the vocalisations made by the puma.
Note: 1 Wild Cats of the World (Sunquists)