The word was first attested in 1938 for the meaning of "looking stupid... because of something you did" and posted in 2007 with that meaning. On August 20, 2002 an anonymous Urban Dictionary user posted the word salty meaning people who seem to be annoyed or angry.
To this generation salt is a way to insult people who are angry but also a way to express your rage. It's a two way street, you could be making fun of someone for being salty one moment and the next you're salty.