What does 'opens at 9 o'clock' mean in Level 13?

It is the level's clock-gated cue: work beneath the timer and wait for the display to reach the opening state before approaching the door.

Do I need a key to open the door?

No key is shown. The door opens through the clock/timer condition, after which a normal rightward walk completes the level.

Mr Potato level guide

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Work beneath the clock until the 9 o'clock opening condition clears the chain, then walk into the right-hand door.

Layout: The room has one flat floor, a large digital clock/timer display in the center, a question-mark marker beneath it, and a chained door at the far right.
Goal: Satisfy the clock-gated opening condition and enter the arched door on the right.
Opening: Move right beneath the clock and use the visible jump around the lower question-mark marker while the display advances.
Danger Zone: The door remains chained during the waiting phase, so walking to it before the opening cue makes no progress.
Mechanics: The exit is controlled by a timed clock/display state rather than by a key or a platform route.

Stay under the clock long enough to see the opening state; do not treat the first chained-door sighting as the finish.
Use the question-mark marker below the display to keep the jump in the intended interaction area.
If the board feels stuck, look for the color with the cleanest path and use that to regain space.

Move right from the starting floor until the potato is beneath the large clock/timer display.
Jump beneath the display and the lower question-mark marker, then settle back onto the floor.
Remain in the clock area while the display advances toward the '9 o'clock' cue.
Move right only after the chain has cleared from the arched exit door.
Continue into the open door to complete the level.

Walking to the right-hand door immediately leaves the potato facing a still-chained exit.
Jumping far away from the clock marker abandons the visible interaction area before the timed state is ready.

What does 'opens at 9 o'clock' mean in Level 13?
It is the level's clock-gated cue: work beneath the timer and wait for the display to reach the opening state before approaching the door.
Do I need a key to open the door?
No key is shown. The door opens through the clock/timer condition, after which a normal rightward walk completes the level.