Where is this phrase originated? What does it mean?

Well it's origin was from the ancient Chinese Taoist. The term "Taoist" will be, hereinafter, referred as the Chinese Taoist. It is the highest realm to be acquired from Chi Kung practice. Before we move any further, we must quote the original Taoist meaning of Chi Kung. Let's say two thousand years ago or little bit longer. Chi Kung is the ultimate breathing method was discovered the Taoist. However, the term Chi Kung was not called by the Chinese Taoist at the time. FYI It was called 吐吶(tu na). It has other meanings too. I will not go into that. I just go over this esoteric term defined by the Taoist.
  吐(tu) is to breathe with the abdomen fully contracted inward.
吶(na) is to breathe with the abdomen fully expanded outward.

Nowadays, Chi Kung is used to replace the term tu-na. So, please disregard what you had read in the past about Chi Kung, with an open mind, for discussion sake to avoid obfuscation and confusion.
Here comes the origin of the phrase "sink chi to the dantien".

Let's start with the term 丹田 "dantien".

The Taoist defined the upper body with three parts called dantien. The upper dantien is from the top of the head to the bottom of the neck. The middle datien is from the bottom of the neck to the bottom of the chest; and the lower dantien is from the bottom of the chest to the bottom of the abdomen.


Dantien(丹田) in the term "sink chi to the datien" is understood referred as the lower datien.

People forget how to breathe as they get older. The breath cannot reach to the middle or lower dantien. Thus they have breathing problem. A most healthy person can breathe down to the middle dantien was not consider to be a problem. However, a Taoist want the breath to reach the bottom datien.

After a long period of beathing practice, finally, the breath has reached down to the lower dantien. The Taoist described as 氣沈丹田(Chi Sink to the Dantien).

From here on, the idea of alchemy comes into play. In addition, the connect the two major meridians together and cosmic orbits are also derived from the idea of "chi sink to the dantien".  That would be covered by another thread.


PS
I want thank starjumper and dedicate this thread to him for his contribution with his famous quote.
starjumper: "Chi and air are/were the same word "