This sounds very much like the short story "The Engine at Heartspring's Center" by Roger Zelazny, first published in 1974. It is also sometimes published under the title "The Engine at Heartspring's Centre".
Let's check your memories against the story's details:
- Enormous, sleek, silver ship: The story's ship is named Silverside. The protagonist describes it as "a great silver fish... a mile long and a quarter-mile in diameter." This matches your description perfectly.
- Traveling for a very long time, purpose forgotten: The protagonist, Charles Eliot Borkman, is the last of a long line of human "riders" or custodians. He has been on the ship for centuries, and the original purpose of its journey is lost to history. He is essentially just a caretaker for the ship's automated functions.
- Thoughtlessly destroys planets: This is the key plot point. Silverside doesn't use weapons. Its method of FTL travel itself is what is destructive. As it passes through a star system, its drive creates gravitational and other spatio-temporal distortions that are catastrophic to any orbiting planets, effectively destroying them and their biospheres. The destruction is a "thoughtless" byproduct of its travel, not a malicious act.
- A victim manifests on the ship: This is the central event of the story. As Silverside is about to pass through a solar system and destroy an inhabited world, a native of that planet, a powerful telepath, projects its consciousness onto the bridge.
- Mechanics of the manifestation: The alien's appearance is a telepathic projection, which aligns with your uncertainty about whether it was "spiritual, interdimensional, an AI." The alien being appears before Borkman and they have a conversation, with the alien pleading for its world and trying to understand the nature of the destructive entity it has encountered.
The story is a poignant exploration of purpose, responsibility, and the vast, impersonal nature of the universe. It is frequently collected in Zelazny anthologies, such as The Last Defender of Camelot.
