why is the death of the first spider so important to Bilbo?

Bilbo does not fit the archetype of the epic hero on a quest, because even the medieval Christianized revisions produce questing hero who are more martial than Bilbo is. It is significant that Bilbo names his sword and that he names it: "Sting." The killing of the spider validates Bilbo into the status of a classic fearless warrior: a status that he had lacked before.