It is my understanding that the term has military origins. A red zone in military terms is the danger zone, generally close to the enemy (red having been a symbol for danger for a long time). The term is analogous to sport as the last 20/22 is where you are more likely to suffer the danger of conceding to the opponent.
I see that Merriam Webster has the first usage in sport at 1983 but the military/danger use predates that somewhat. Indeed, there is the 1966 film, "Red Zone Cuba", which is centered around a military facility.
During World War I, the decimated areas of France were known as the Red Zone.
To sum up my answer, the term "Red Zone" has been used in various contexts as a term for danger. Its application in sport is to represent the danger of your enemy and it was first coined by ex-Redskins coach Joe Gibbs in 1982.
The term "red zone" was coined by then Washington Redskins coach Joe Gibbs in a 1982 interview with the Washington Post. Gibbs used the term to describe the heightened sense of urgency on both sides of the ball when the offense gets close to scoring. He called efficiency in the red zone an important part of that team's success.
From Wikipedia (as linked by Orangecrush)
The area is not literally colored red and the term is used mainly for statistical purposes
As far as I can tell, the etymology is that they needed a name, and "red zone" sounded cool. Nothing particularly special or notable.
From here, and the description of the zone, red zone(s) are the vital area for scoring a touchdown (from most rushes and passes), and it also the nice range for the field goal kick, too. Danger can be detected by the defense, so it seems legit to name that area the "Red Zone".