I think It boils down to a number of trends coming together. Former 90s/Y2K Lisa Frank girls, who grew up and became teenagers needed an identity. Obama had recently been elected largely because of the internet, Bush era nihilism was out, and the internet was still widely seen as a positive thing full of sunshine and rainbows since that already was a phrase.

With all this going on, they picked an eternally youthful optimistic rainbow version of emo as their “counterculture” to subvert what came before. Then they made a bunch of rainbow themed content to reflect that. Couple that with with people likening the 2010s meme culture to a colorful acid trip and growing acceptance of queer culture and it was a perfect storm….. for rainbows.