Let me begin with a little bit of history. James Prescott Joule did experiments along the following lines. He would have a calorimeter, with paddles inside so that moving the paddles would churn the water inside, and the friction of that churning would raise the temperature of the water. One of the ways of driving the paddles would be to have a weigth suspended on a string, and the string running over a pulley so that the weight descending would drive the paddles.
As we know: the outcome of those experiments was that the temperature rise of the water is proportional to the height over which the weight descended.
Let's say that Joule tried two different heights, say 1 meter height and 4 meters of height. (Yeah, you see a squaring coming)
4 times the height gives 4 times the amount of energy transfer to the water in the calorimeter.
Also, Joule verified that the amount of energy transfer is independent of how fast or how slow the weight descends. The weight can drop fast or slow, at the end the same amount of energy is transferred.
To find a relation with velocity.
To find a relation with velocity we compare the case of a weight not suspended, but free falling.
If an object is allowed to free fall it gets to keep all velocity it gains. Presumably in the course of free fall the change of energy is the same as in the churning paddles case.
How much velocity does a weight have after 1 meter and 4 meters of dropping?
For simplicity I set the acceleration to 2 m/$s^2$
With an acceleration of 2 m/$s^2$ after 1 second the distance traveled is 1 meter, and after 2 seconds the distance traveled is 4 meters. That is a quadratic relation
How do the velocities compare after 1 and 2 seconds of time?
As we know with uniform acceleration velocity increases linear with time.
With an acceleration of 2 m/$s^2$ after 1 second the velocity is 2 m/s, and after 2 seconds the velocity is 4 m/s.
We assume that such a thing as kinetic energy exists. Can we express this kinetic energy in terms of mass and velocity?
It follows logically that if you express kinetic energy in terms of velocity the kinetic energy is proportional to the square of the velocity.
Why is the kinetic energy proportional to the SQUARE of the velocity?
Here is one way of looking at that:
Let's say you shoot a marble into a lump of clay. The amount of damage that the marble does is proportional to the depth that it penetrates into the clay.
Assume that the rate of deceleration is constant. For simplicity let's say that it takes 2 units of time for the marble to travel into the lump of clay and come to a stop. With constant rate of deceleration 3/4 of the penetration distance was travelled in the first unit of time, and 1/4 of the penetration distance was travelled in the second unit of time. At the same time: the marble lost 1/2 of its velocity in the first unit of time, and lost the remaining 1/2 of its velocity in the second unit of time.
What this shows:
when you go fast you travel more distance, so you do more damage, in the same amount of time. Your kinetic energy is the amount of damage that you will do on impact, therefore kinetic energy is proportional to the square of your velocity.