Become angry; lose self-control.
The colour red has many associations - heat, heated emotions and violence, communism, a sign of warning (as in traffic lights etc), ripeness (in fruit etc), the colour representing the British Empire on maps and, of course, blood.
It is widely thought that 'see red' derives from the sport of bull-fighting and the toreador's use of a red cape to deceive the bull.
The phrase is known from the early 20th century and so is easily predated by the ancient sport, and more to the point, the knowledge of bull-fighting parlance in English-speaking countries, which dates from the mid-18th century.
That proposed derivation is backed up by the existence of the earlier phrase - 'like a red rag to a bull'. This is found in Charlotte Mary Yonge's novel The pillars of the house, 1873:
"Jack will do for himself if he tells Wilmet her eyes are violet; it is like a red rag to a bull."
Bulls can't actually see in colour and are attracted by the waving of the cloth rather than the redness. That doesn't detract from the red cape theory as the origin of this phrase however.
There is an alternative possible derivation. The phrase may be an adaptation of an earlier American expression - 'to see things red'. That is unconnected to bull-fighting and alludes to a state of heightened emotion when the blood rises and we become angry - what these days is more often referred to as 'the red mist'. The earliest known example of this is in Jerome K. Jerome's Three men on the bummel, 1900:
"I began, as the American expression is, to see things red."
'See red' itself is found in print the following year. Lucas Malet, the pseudonym of Charles Kingsley's daughter Mary St. Leger Harrison, wrote the romance The history of Sir Richard Calmady in 1901, which included this line:
Happily violence is shortlived, only for a very little while do even the gentlest persons 'see red'.