Silver is supposed to react against Vampires and Werewolves because of it's association with the moon. Ancient mythology said that silver came from the goddess of the moon and contains all the purity of the goddess. (Diana and Athena were the Roman and Greek goddesses of the moon and were considered virgins, huntresses and protectoresses)
Werewolves are controlled by the energies of the waxing and waning of the moon, being that they cannot control their change when the full moon rises. Because they become beasts of violence, creatures who hunt the helpless, they insult the goddess. The goddess sends silver to man to protect them from the creatures who hunt them. Anyone wearing silver is safe because they cannot touch them.
Presumably, this would apply to Vampires as well, as Vampire stories and Werewolf stories are inextricable. There is the strong element of the Christian influence as well (more so than with Werewolves), as silver is associated with the Virgin Mary (the woman clothed with the sun and the moon under her feet, associating her with the goddess of the moon) So wearing silver, especially in the form of a religious medallion becomes a controlling factor in the Vampire mythos concerning silver.
In the Sookieverse of Bon Temps, Vampires exhibit an allergy to silver where they are weakened and burned by silver and whereas in mythology the werewolf can only be killed by silver, the weres and shifters can be killed with plain old bullets.
Sources:
The Vampire in Lore and Legend by Montague Summers
The Werewolf in Lore and Legend by Montague Summers
The Malleus Malficarum by Kramer and Sprenger
Written by Aslinn Dhan