Arachne is a maiden who lives in an obscure little town. She was famous there because she's a well-skilled weaver.
One time, she can't take the people murmuring that her skill was given by Athene so she shouted that there's no one better than her. An old woman from the crowd disagree with her that's why she challenged Athene for weaving competition.
Surprisingly, the old woman was Athene as she transformed her true self. As they compete, Athene weaved her heroic events when she fought Poseidon.
Arachne weaved evil and unworthy actions of gods that makes Athene to be insulted. She burst her anger by destroying Arachne's loom and struck her across the face.
Athene then turned Arachne into a spider.
All spider was descend from Arachne, that everytime people will see one of it, it reminds the competition happened.