Why is horse serum used for snake bites?

Simply because a horse is a large animal, who's immune system can cope with being injected with small doses of venom. The amount injected is no-where near enough to cause the animal to suffer - but enough to 'trick' the horse's immune system to produce anti-bodies to fight the toxins.

The horse's blood is 'harvested' after allowing the immune system sufficient time to produce large quantities of anti-bodies. The fluid is then spun in a centrifuge to separate out the different blood components. The anti-bodies are kept - and 'grown' in a laboratory to make large quantities of anti-venin..