The project that Scott Manley is referring to was known as Project A119, and was run at the Armour Research Foundation (ARF), which was based at the Illinois Institute of Technology. The ARF is now known as the IIT Research Institute.
The official title of Project A119 was A Study of Lunar Research Flights, and Volume 1 of the report, produced in 1959 by the Air Force Special Weapons Center (now known as the Air Force Nuclear Weapons Center), can be read or downloaded as a pdf on archive.org.
Sagan was a member of a team of up to ten people led by Leonard Reiffel. This article published in the Guardian newspaper on 14 May 2000 notes that:
Sagan's biographer, Keay Davidson, discovered that he had disclosed details of it when he applied for the prestigious Miller Institute graduate fellowship to Berkeley.
Davidson's biography, titled Carl Sagan: a Life, is also available to read on archive.org. The incident mentioned in the Guardian article is described on page 110:
In the fierce competition for the Miller Fellowship, Sagan needed an ace, something truly distinctive, something that would make the Miller judges sit up and take notice. So he decided to confide to them information that he was required by federal law to keep secret. He revealed his research at the Armour Research Foundation on the remote detection of lunar nuclear explosions. He must have known the risk he was taking. The information was classified; he had previously cautioned Muller not to discuss it with others.
So it would seem that the revelation was not actually an accident, as stated in Manley's video, nor was it quite as "public" as implied.
In the event, Sagan's application was successful. His two-year fellowship started in September 1960 and brought with it a stipend of $7,500 per year and a $500-a-year "contingency fund for travel, supplies, and equipment".