KnifeMissile

I don't know if you will call this "clever" but it is an intuitive explanation (rather than a mathematical derivation).

First, mass is defined as the measure of one's inertia. In other words, when we talk about the mass of an object, we are really talking about how difficult it is to move (accelerate) it.
Now, obviously, the more kinetic energy you put into an object, the faster it will go. However, as it turns out, the faster something is moving, the harder it is to move it! I mean, as the speed of an object gets closer to the speed of light, the harder it becomes to move it any faster. For example, an object that's travelling at 0.9c will have a certian amount of kinetic energy. If we were to double the amount of this energy, we will not have doubled it's speed and, in fact, it's speed will only increase to 0.96c! Why is that? In a sense, it is because the object is now harder to move and, it is in this sense, we say it's mass has increased.

Does the kinetic energy "turn into" mass? It depends on what you mean. It's not like a heat engine, where heat energy gets turned into kinetic energy. It's more that the increased mass is an aspect of the increased kinetic energy in the object.

I think there are two interesting things to note, here.
First, energy is relative. A bullet has a lot of kinetic energy relative to the shooter but, relative to another bullet, say, the first bullet has no kinetic energy! Mass is also relative in exactly the same way...
Secondly, a mass has energy even if it is not moving! Of course, everyone knows this but I think it has now been given the proper context in which this can seem like an interesting idea, rather than just something scientists say...

Oh, and one last thing. Your post was a little confusing 'cause you ask "does the kinetic energy [formula] partly turn into energy," but kinetic energy is energy! So, what did you mean?