Reading that headline should tell you everything you need to know, and make you question exactly what was going on in the minds of the producers, but it happened, and we're doing to look at it in more detail. These days, if you wanted a red horse, you would just shoot your scenes with a regular horse and let CGI do its thing to make it red. There was no CGI in 1939, and they really wanted that horse to be red.
The best way to achieve the effect was to coat the horse - which was naturally white - with gelatin powder. The horse apparently didn't mind too much and was quite fond of licking the powder. You can actually see the cab driver trying to prevent the horse from licking itself. That wouldn't be too bad, until you remember that gelatin back in the era the film was made was often derived from horses. So, you're actually watching horse cannibalism.