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..