UPDATE: As several people have pointed out, this is actually not a "lost" scene from the "In Thy Name" Star Trek: Phase II pilot from 1977 (despite the description on the YouTube video page) but a scene from the 1974 movie Pray For The Widcats. The yellow and red shirts do look very Trek-esque though which probably confused matters for whoever originally posted the video. The rest of the information on this page about Star Trek: Phase II (1977) is correct though.
Here's a scene from the unaired TV pilot for Star Trek: Phase II in 1977 showing William Shatner reprising his role as Captain Kirk, who is now married with children and having an existential moment. Robert Reed (yes, him from The Brady Bunch) as his first mate. Leonard Nimoy did not sign up for Phase II.
And here's some CG animated shots of the never-aired Phase II U.S.S. Enterprise. Obviously, the graphics are recent, but based off the ship design for the then proposed new series.
And here's a short documentary on the Phase II - The Lost Enterprise...
There's more information on the Phase II series here including the premise for the show and new characters. There was to be a 2-hour pilot and 12 additional episodes. Star Trek: The Motion Picture was eventually made instead.
Filed under: Star Trek: Phase II