This is a very questionable usage of the term "self-fulfilling prophecy." A self-fulfilling prophecy is a prediction that comes true simply because the prediction was made. As an example, suppose Warren Buffet were to declare that the stock price of Coca-Cola was going to skyrocket. Since Warren Buffet is respected as a financial analyst, thousands of people would be motivated to buy Coca-Cola stock, thereby raising its price. The simple act of Buffet predicting the price increase would directly lead to its actual increase.
The author of the paragraph you've provided is implying that a primary factor in climate change is pessimism. If the public were to stop believing that climate change is an inevitable terror that awaits us in the future, then we could actually address the problem. "We can dispatch the self-fulfilling prophesy that we're cooked" can then be translated as "We can get rid of this fear of catastrophic global warming that only allows the problem to proliferate."
Personally, I'm not convinced that this actually qualifies as a self-fulfilling prophecy. I don't see an evidence that fear of a bleak future is nearly as powerful an underlying factor in climate change as greed and arrogance.