What is the origin of the term "ginger" for red-headed people?

The OED online has this listed as definition 5.B of "ginger":

B. adj.1 dial.

Of hair: Having the colour of ginger. Of a person: Sandy-haired. Of a cock: Having red plumage.

a1825 R. Forby Vocab. E. Anglia (1830) , Ginger, of a pale red colour, particularly applied to hair.

1834 T. Medwin Angler in Wales I. 35, I perceive a fine red or ginger game-cock in the yard.

1886 R. Holland Gloss. Words County of Chester, Ginger, sandy-haired. ‘He's a bit ginger.’

1897 Daily News 10 Sept. 2/6, Complexion and hair brown, moustache ginger.

So, as indicated by "having the color of ginger," someone at some point was quite convinced that the color of ginger was indeed red.

I am only familiar with ginger roots that do not possess any red coloring.

Contrary to many cooks' experience, ginger root (Zingiber officinale) is not the only ginger plant. The Red Ginger plant of Malaysia, Alpinia purpurata, is a brilliant red:

Circumstantially, it was in the late 18th century and the 19th century, during which ginger came to mean “red-haired” or “red-plumed” according to the OED citation in Amanda's answer, that the British occupied parts of Malaysia (among other countries), and started coming into regular contact with the Red Ginger.