In horse racing, "nose" is one of several ways to describe a distance shorter than a length, which is the length of a horse. The winner of the race is said to win by a certain distance, which describes the position of the horse in second place when the winning horse reaches the finish line. If the second horse's nose is even with the winning horse's tail, the horse has won "by a length." If the race is closer, the distance could be half a length, a neck, a head, or -- closest of all -- a nose.
An announcer describing a race, for example for a radio broadcast, will describe the horses' positions relative to each other and to the track, often using "and it's..." to indicate the ultimate outcome of the race.
What does "fat ass by a nose" mean here?
As others have noted, it's Fat Ass by a nose means that Fat Ass has won the race, but just barely.
Now I looked for some videos to illustrate this, but had some trouble finding them; I guess that as television broadcasting of races has become more common there has been reduced demand for the announcer to describe the distances. In 1973, however, the word "length" is in full display, along with (uncontracted) "it is...," as Secretariat wins the Preakness.
See also Who Won? (1923) and The 2016 Belmont Stakes - Creator wins by a nose for "by a nose."