Points on the cylinder can be used to represent all possible states of our pendulum, for example.
The "lengthwise" position of the point represents the speed and direction of the motion of the pendulum.
The position of the point in the "circular" direction represents the angle of the pendulum.
We usually trace a line along the path followed by the moving point (as in the picture) to make it it is easier to follow what is happening.
Of course, the way in which the point moves will be determined by the rules which govern the system.
QUESTION: Can you work out how the penulum would have to move for its state to be changing as shown in the animation? Does this seem possible?
Visualising the changing states of a system as a point moving through space helps our intuition about what is possible, even though there are often too many measurements to allow a realistic picture to be drawn.