A spider got caught in a radiation test, the radiation was slowly killing it. Before it died it bit a student by the name of Peter Parker. The spider's bite caused changes to Peter's DNA that gave him certain powers of a spider such as super strength and agility, the ability to climb on any surface, shoot spider-webs using devices of his own invention called "web-shooters," and react to danger quickly with his "spider-sense", enabling him to combat his enemies.
Also attending the exhibit was a student by the name of Carl King, who witnessed Parker being bitten and connected it with the later emergence of Spider-Man. Seeking to gain the same abilities as Peter, King decided to eat the spider, only for a completely different result to occur. His body began to break down over time, eventually metamorphosing him into a swarm of 1000 spiders sharing a single consciousness.
Somehow the spider had time to bite another person inbetween those events, Silk, who got the same abilities as Peter Parker, only better, for some reason.