How speed affects time? Rewrote the question so hopefully it makes sense. 
What causes time dilation? Is it due to the slowing of the atoms vibration that effects the measurement of time?
If we are using the vibrations of atoms to measure time with an atomic clock.
An atom has electrons that are vibrating around the nucleus at the speed of light.
While the atom travels near the speed of light, the surrounding elections cannot exceed the speed of light, so when they move around the nucleus in the same direction that the atom is traveling it takes longer to vibrate in that direction, reducing the frequency of the vibrations.
 A: The best and simplest way to view it, in my opinion, is to imagine you are always traveling through spacetime, not space and not time, at the constant velocity of the speed of light. If you increase your spatial speed, relative to another observer, then you decrease in time speed, again relative to another observer.  You won't notice any difference, that's why I keep referring to another observer.
Is that not simpler to imagine, and also  closer to the true idea of time dilation, than your analogy?  
As to the cause  of time dilation.....it is, to the best of my knowledge, impossible to define or explain  time itself in any way , other than the operational, pragmatic and realistically only way that physics can deal with time, that is time is what a clock measures.
Your revision seems to deal with the measurement of time, rather than the cause of it. Apologies if I misunderstood you.
It seems you can get a lot of philosophy based ideas on this, which you might find interesting, or you might find physiological based arguments, but it's not physics in either case as we can't apply physical reasoning.  There is no math, no experimental techniques and no predictions in these   discussions of time, again as far as I know. 
If you read this article : Entropy and the Second Law of Thermodynamics, you might find a slight twist in the way we view time, as a probability based "thing", rather than cast in stone as regards direction. Apologies if you are already familiar with it.
A: Analogies are always limited, but I really don't like yours because it evokes the idea of a "drag" or "viscosity" of spacetime, which, to my mind, bears little likeness to what we know about time dilation.
Actually it's a great deal simpler, but subtler, than your analogy. Time dilation is quite naturally implied by basic, everyday-observable (or at least reasonable) symmetries of the Universe, as I describe in my answer here. Grounded on symmetries, the idea is unfortunately more abstract and not really analogous to other, everyday physical principles like fluid drag.
TáMéCeart's Answer is a good way to think about the geometric rule that governs the transformation, although one has to understand that the geometry is different from Euclidean geometry.
