There is a very plausible explanation here.
Basically, it explains that meat colour is due to the protein myoglobin (a haem-containing protein related to haemoglobin). There are two types of skeletal muscle: fast-twitch and slow-twitch (Wikipedia). Slow-twitch muscle is red muscle because it contains lots of myoglobin. Fast-twitch muscle is white muscle, containing less myoglobin. The Wikipedia link explains a little more about these muscle types and their functional differences. Evidently the lifestyle of the ostrich dictates that it have more fast-twitch muscle, and so it has red meat. It isn't unique in this: duck and goose are classified as red meat too.
Incidentally the role of myoglobin is to improve the rate of diffusion of oxygen from the haemoglobin in the capillaries through the cytoplasm of the myocytes to the mitochondria. It has also been suggested that it acts as an oxygen storage mechanism in diving mammals which have very high levels of myoglobin in their muscles, but this storage function is unlikely to be important in other mammals, despite what you may read in textbooks.