First, The Culture's spaceships are enormous. The largest type we encounter, the General Systems Vehicle, is 200 km long and can house up to 6 billion people; while these serve as habitats for a civilian population, these are still spaceships, capable of moving at great speed.
Secondly, the ships have no physical hull. Instead, their structure is maintained by field manipulation. Banks doesn't go deeper into how this works, but it's clear The Culture has technology to manipulate physical reality similar to classic science fiction "force fields" that allows ships to maintain an atmosphere and protect against physical damage. Notably, in several books, the ships modify both their interior and exterior structure while traveling in order to optimize themselves for some purpose.
Thirdly, an important part of The Culture is that the ships are, in a sense, alive. The Minds, which are the AIs that control them are largely inseparable from the ships they inhabit. Clearly we've had AI-controlled ships before (HAL, Alien's Mother, and so on), but these have always been subservient to humans. With The Culture, a human boarding a ship is a guest of the Mind, and ships don't have captains or comamnders. The only other author I know about who has done anything similar is Anne Leckie.