The thing is, it's not semantically nonsensical. If you examine it with a bit of leeway, it's a perfectly normal sentence.
Colorless: dull, uninteresting. Flat, without energy.
Green: New, fresh, untested, unproven in the field.
Sleep: Lie dormant, paused, hibernating.
Furiously: With great energy, restlessness.
So it follows that a colorless green idea is a new, untried idea that is without vividness, dull and unexciting...To sleep furiously may seem a puzzling turn of phrase but one reflects that the mind in sleep often indeed moves furiously with ideas and images flickering in and out.
So what's the meaning of all this? That new ideas, not yet sharply defined, circulate in the unconscious, rapidly altering at a furious rate.
Hooray for the English language.