What happens to the light you see as a far away object approaches you with constant velocity? A rocket ship 1000 light years away from me approaches me with a constant velocity. At first the light i see is 1000 years old and the ship is closer than it appears. Later the difference between the actual location of the ship and where i see it has been reduced until the ship reaches me and the difference is negligible. I am confused by what mechanism this occurs and how one might perceive this.  
 A: The difference $\Delta x = x-x' $ between the position $x$ of the rocket when the light was emitted and its position $x'$ when you receive the light, is the speed of the rocket $v$ times the time $t=\frac{x}{c}$ taken for the light to reach you from the initial position of the rocket :  $$\Delta x = vt = \frac{vx}{c}$$ As the rocket gets closer ($x$ decreases) the time $t$ taken for the light to reach you gets smaller, while the speed of the rocket remains the same, so the discrepancy in distance $\Delta x$ gets smaller also.
This happens because the speed of light $c$ is not infinite. It is very very big, but the enormous speed of light can be offset if either $v$ or $x$ is very big also.
The same effect happened when armies or navies moved around the world more than 150 years ago. The only way of communicating with them was by fast sail boats or relays or horse-riders going overland. It took several days for messages from the army or navy to reach their destination at their country's capital on the other side of the world. During this time the army or navy, travelling slower, had moved to a new position.
A: 
I am confused by what mechanism this occurs and how one might perceive
  this.

Assume for concreteness that the rocket ship is moving towards you with speed $0.5c$ and that the rocket is $1000$ light-years away when $t = 0$.  You observe (with your rods and synchronized clocks) the distance (in light-years) of the rocket to be
$$d(t) = 1000 - 0.5\,t$$
where $t$ is measured in years.  Let $\tau(d)$ by the time it takes for you receive light emitted by the rocket when it was at distance $d$
$$\tau(d) = \frac{d}{c} = \frac{1000 - 0.5 t}{c}$$
where $c$, the speed of light, is one light-year per year.
See that the rocket arrives at your location when $t = 2000$ years while you receive the first light from the rocket when $t = 1000$ years.
Thus, by the light you receive from the rocket, the 2000 year trip is compressed into 1000 years.
For example, assume that there are 'mile-posts' every light-year from you to the rocket and that the rocket sends a pulse of light each time it passes a light-year marker.
You observe (with your rods and synchronized clocks) that the rocket takes 2 years to travel between light-year markers but, beginning at $t = 1000$ years, just one year elapses between the received pulses of light.
