The parts responsible are the clutch and cam.
As the key bottoms out, besides aligning tumblers, it engages the clutch. As result, as you turn the key (and the core) - the cam turns too, moving the rest of the lock mechanics.
With dual-core locks (openable with a key from either side) the clutch is just friction-based, so once a key pushes it into "engaged" position, it stays in it until pushed to the position engaged for the other side once a key from the other side is inserted. For locks that just have a simple knob for opening from inside, the clutch is actuated with a spring - in absence of a key it engages the knob side and the core (key side) is disengaged.
As a side effect, it's a slight improvement of lock safety - when picking such lock, even if all tumblers are latched, all you achieve is turning the core uselessly while the deadbolt doesn't move. You need to depress the clutch to engage the cam and transfer the rotation to the rest of the mechanism.
Also, if you want to lock someone in, with the lock only lockable from your side, but openable with the knob from the other, after locking it, insert a wrong key. This way the clutch engages for the key side, and the tumblers block the core - the lock can't be opened from any side.