You can slice up the cylinder of say height 100 cm into equal parts of 10 cm each. Because the circular surfaces of each part are congruent, the water level will increase or decrease constantly. Hence the height of the water level increases or decreases constantly.

For cone, we similarly slice up the cone into equal heights each. This time, the size of the circular surfaces decrease as the height gets lower (that's the design of the cone).

When the water is filled in, naturally the water level rises quickly at first, but then eventually it slows down as the height increases. The shape of the graph is something like the left half of the graph y = -x^2.

When the water flows out (from the bottom), at first the height will decrease slowly from the top (for the same reason as explained previously. As the height gets lower, the height reduces very quickly. The shape of the graph os something like the right half of the graph y = -x^2.

You can think of this like filling up your home sink fully with water at a constant flow rate and watch how the water level increases or decreases.

When water flows into the vertex of an inverted cone, the water level increases slowly, but gains speed as the height increases. The shape of the graph is something like the right half of the graph y = x^2.