Following the outbreak of war with the Klingons in 2256, the Glenn and the Discovery were tasked with conducting experiments on a new type of faster-than-light drive utilizing spores from the fungus Prototaxites stellaviatori. (DIS: "Context Is for Kings")
In contrast to Discovery, which cultivated a mushroom forest, the Glenn team, led by astromycologist Straal, used dry spores for their spore drive. This led to a tardigrade animal infiltrating the ship's cargo stores. Finding that the animal could communicate with the spores, the crew built an invasive mechanism in their reaction cube that would connect the animal to the drive system, performing a role akin to a supercomputer. This allowed them full navigational control over the probabilistic nature of the spore drive, enabling them to make jumps of ninety light years to the Beta Quadrant in 1.3 seconds, but caused the animal intense distress. The crew added additional shielding to their engineering lab to contain the animal.
Eventually, however, one experimental jump led to a catastrophic accident, when the Glenn hit an undetectable Hawking radiation firewall while exiting the mycelial plane. The resulting impact caused the drive system to "spin out", exerting extreme helical torsion that killed the ship's complement and left evidence on the hull of a basidiosac rupture. The animal, however, was apparently unaffected and broke out of the reaction cube and engineering. A group of Klingons then boarded the Glenn, only to be killed by the animal, which had become highly aggressive.
The Discovery, assigned to investigate the accident, deployed an away team to the Glenn. The team was able to salvage logs and key technology from the Glenn, though security officer Kowski was killed by the animal. After the away team evacuated, the Glenn was scuttled via two photon torpedoes. However, the animal had secretly been transported to the Discovery by Captain Gabriel Lorca's orders. Lorca was impressed by the animal's metal-tearing claws and resistance to phaser fire and bat'leths, and hoped it could be weaponized. (DIS: "Context Is for Kings", "The Butcher's Knife Cares Not for the Lamb's Cry")
The Glenn was presumably named after John Glenn, the first American to achieve orbit around the Earth. He also became the oldest man in space when he returned in 1998 aboard the Space Shuttle Discovery.
The set used for the Glenn's interior was a reused set that, with just a few changes, had previously served as the interiors of the USS Shenzhou and went on to be used as the innards of the USS Discovery. To portray the Glenn, the production crew modified the set by adding scratched walls, busted doors, flickering lights, dangling wires, and lots of blood. (AT: "Context Is for Kings")
The Glenn appears in the "Age of Discovery" expansion to Star Trek Online. Under the command of Captain Deb Underwood, the Glenn comes to the aid of the player character while defending Delta Volanis Station from a Klingon attack. The player character's crew then goes aboard the Glenn when she's boarded by Klingons led by J'Ula, sister of T'Kuvma, who seeks to capture the ship's mycelial technology. This also explains the bodies of the Klingon boarders killed by Ripper when the Glenn is found by the Discovery. The Glenn class, a 25th century upgrade of the Crossfield class added in the game's tenth anniversary "Legendary Starship Bundle", is named for her.