Why and how was Arachne punished, and what lesson are readers supposed to learn from this punishment?

she got turned into a spider by Athena for being awesome

Explanation:

Arachne was a weaver and a really good one. At some point, she started getting audiences that would watch her weave. We don't really know how Athena got into this, maybe she was called, maybe she saw Arachne being awesome, but we know that Arachne was proud enough that she wasn't going to give the credit of her amazing weaving to Athena. Athena comes down and basically issues a challenge to see who's the better weaver. Arachne and Athena go to work, and when they're done, Arachne's piece ridicules the gods, saying how stup!d they were, and how amazing nature was in comparison, Athena's work praised them and said how awesome industrialism and architecture was. Some stories say Arachne won, some say Athena did, and some say it was a tie, but they all say that Arachne made a vital error, she insulted Athena, and by this time, Athena was having enough with this Arachne. So she beat her up, and hit her with the weaving thingy, and then turned her into a spider, so she could weave forever. Nice story right? So the moral? Don't be vain? Don't stand up to gods? Be nice to people with more power than you? Remember your place maybe? I don't know, you decide.

i would say it means that you're never alone, that there will always be people supporting me and showing me that they care for me and love me

explanation:

yes, they are. i think.