This paragraph is worded a bit confusingly, but it was pretty clear to me: the fifth freedom is the fifth of nine aviation freedoms from a treaty signed in 1944.
Airline 1 from Country A
Airline 2 from Country B
Airline 1 is allowed to fly from A to B, refuel in B, and then fly from B to C. But the fifth freedom allows Airline 1 to pick up people in country B and provide them cheap flights to country C, thus competing with Airline 2, which is native to country B.
It could be entirely possible that my understanding is incorrect, though... but that's what I got from their explanation.
I do completely agree with other posters that this is thinly-veiled advertising, though. This stands out with their Japan flights for me, since I live in Japan and occasionally look for tickets.
It's interesting that they point the price for Osaka-Honolulu because it's surprisingly low (~80% cheaper), but they conveniently neglect to mention that the Tokyo-LA flight is only 10-15% cheaper than a domestic airline.