They don't need to be on different channels, but it's recommended for performance (due to co-channel interference).
Think of it this way: Each channel provides a fixed amount of bandwidth, and neighboring APs need to overlap slightly to provide seamless coverage. As a first order approximation, if two APs are on the same channel you'll have twice the range, but half the bandwidth because the spectrum would be shared between them.
(It's actually slightly worse than this, because collisions will happen causing additional overhead.)
Think of it this way: Each channel provides a fixed amount of bandwidth, and neighboring APs need to overlap slightly to provide seamless coverage. As a first order approximation, if two APs are on the same channel you'll have twice the range, but half the bandwidth because the spectrum would be shared between them.
(It's actually slightly worse than this, because collisions will happen causing additional overhead.)