It uses the same adaptive immunity that mammals (us included) use to develop antibodies for diseases. A key bit of information that seems to be missing from all the answers here is that most venoms are proteins, which is what mammalian immune systems usually produce antibodies against. (Not that we can only develop antibodies against proteins in particular - the key metric is the size of the molecule. Larger molecules have - by virtue of being large and structurally diverse - more unique structures than smaller ones, so chemical interactions between large molecules can be more specific and therefore stronger than those between small molecules.)

So armed with the knowledge that all or most mammals have similar immune systems that can develop antibodies against virtually any protein, and that venoms are proteins, it stands to reason that you can make antibodies to venom in most mammals. We use mammals like goats, rabbits, sheep, and horses to make other antibodies for scientific research, too.

Edit: A bit of a primer on poisons might be helpful here. As said above, we can develop antibodies against large poisonous biomolecules, whether they are enzymes that directly interfere with our biochemistry (making us sick), or they are receptors on viral particles that act during one step of a longer process that in the end interferes with our biochemistry (and thus make us sick). But some poisonous molecules are small, so we cannot develop antibodies against them. Arsenic (as arsenate), cyanide, heavy metals (lead, cadmium, radioactive iodine), and mustard gas are examples of such poisons. They act by either displacing small biomolecules (arsenate replaces phosphate; heavy metals replace other metal cofactors like iron, copper, and cobalt), competing for an enzyme's binding site (cyanide outcompetes oxygen), or reacting irreversibly with a biomolecule (mustard gas reacts with DNA). Because they are small, these poisons look a lot like other chemical species that occur frequently in biology (which is precisely why things like arsenic and lead are toxic), so antibodies against them would cause autoimmunity. Venom enzymes, being large, have unique structures that occur nowhere else in the target organism's own biology, and so they can be uniquely identified and bound by antibodies.