In "The Men Who Stare at Goats (2009)", this line appears at the beginning:
The movie looks like fiction, with superpowers etc.
Why does it say that more of this is true?
They say that more of it is true than you would believe, and considering the farfetched material, you'd think that none of it is true... But amazingly, some of it is based in reality. The US military did experiment with the paranormal and it was funded by Congress. ABC News covered it extensively, and here are some choice bits:
Major Paul H. Smith calls it his "Men in Black" moment. It was 1983 and he was working as a Middle East analyst at Fort Meade, Md., when a fellow intelligence officer approached him with a highly-classified, so-called "black project." [...]
For seven years, Smith took part in a congressionally-funded program focused on training officers in "remote viewing," or a paranormal skill that supposedly allows a person to see a target despite the restrictions of space or time. [...]
Smith and others familiar with the actual events upon which the movie and book [it was based on] are based say they only loosely reflect reality. But they also say that in ways both formal and informal, the military did dabble in the paranormal. And, they claim, they had some success.
Even the goat-staring is loosely connected to real events. From the same article:
Col. John B. Alexander, a retired intelligence officer who is on the IRVA's executive board, didn't take part in the remote viewing program, but participated in other activities loosely described in "The Men Who Stare at Goats." [...] The story of the goat stared to death, he said, likely came from casual experiments with the martial technique, dim mak or "death touch." [...]
Though he didn't see someone use the technique on a goat, he said he saw the necropsy report for a goat that had apparently experienced the "death touch." He said across the goat's rib cage, he saw a path that looked just like that of a bullet, but there was no wound of entrance or wound of exit.
The army did research the paranormal for counter-intelligence and military applications. For example the Stargate Project. So, I'm guessing that's the real part, the part hard to believe. Military men, in military suits, did try to "stare goats to death".