When we pour a liquid into a glass-sized container, why do we often start close, then increase the distance during the pour, then decrease just as the container gets full?
Is it to look cool? Or does it actually make it easier to pour?
Well, it’s the angle that really determines it, I believe. And if the containter is too big to get a good pouring angle close to the glass, then we have to move it away. This then carried over to small containers, because it DOES look pretty cool.
Why don’t you try doing it the other way and see what happens?
WAG here. When you first start the pour it’s common to tip the liquid container further than necessary to get things flowing. There’s also some uncertainty of level of liquid in the container. This results in uncertainty as to the velocity at which the liquid comes out. If it comes out slowly, it’ll drop mostly straight down, but if it’s moving fast, it’ll arc horizontally, and likely miss a glass held too far underneath the spout. Hence we hold the glass close at the start of a pour.
Once things are going, it’s nice to be able to see what’s going on and control the flow rate, so we back off a bit. -Nice long arcs of milk falling through the air look pretty too.
At the end of the pour, we tilt the container back up, so the arc of liquid becomes less. However, there’s always the chance of a bit of dribbling, i.e. flow down part of the side of the container, followed by a vertical drop, so we move the glass closer to the container in order to avoid spillage.
Or, we might do it all because of quantum stuff that no one understands 
Cos when you start it’s hard to aim from on high.
Once you start you have the flow of the liquid itself to guide you, so you can lift the source container away from the target, giving you more room to tilt and increase the flow.
As you finish pouring you need to move closer. It’s hard to judge when to stop pouring from a high position if there’s a whole column of liquid still in mid-air that’s yet to arrive at the target.
I have seen bartenders doing this when they are drawing beer from a keg that’s gone nearly flat. When they move the glass away from the tap it forces more carbonation out of the beer to cause a better looking head.
Have dinner at a Morrocan restuarant. At the end of the meal you get a small cup of mint tea and it’s traditional for the server to hold the teapot as far away from the cup as possible to make a show of pouring. Although they do start and finish close to the cup.