As to why people think we wouldn't be able to cope outside the EU, I'd say it comes down to the fact that we currently have very favourable trading arrangements within the EU, and a streamlined process for importing and exporting components, products and services, and it's extremely difficult to see how that might be replicated outside the single market. While some leavers were advocating staying within the single market before the vote (presumably so as not to spook voters) many are now saying that this wouldn't be a "real" Brexit. Some are saying the same about staying in the customs union too.
Modern trade agreements are incredibly complex and detailed, and take many years to complete with very large numbers of experienced (and expensive) negotiators. We don't currently have many of those. If we are lucky and careful (and stop insulting the rest of the EU) we might get a decent transitional agreement to see us through to the other side. Or we might not.
Before the ref a poll of foreign investors showed that for 75% of them, full access to the European single market was a key reason for being in the uk. If we lose that we lose the investment, the jobs, the tax revenue.
Then there is the financial sector. Whatever we may think about their practices, they contribute around 11% of the U.K.'s total tax take (around £60 billion pa if you want to do the comparison with the £350 million eu contribution). If, as seems likely, we lose the bulk of that to France or Germany there is just no way of making that up, so our schools, hospitals, social care etc all suffer.
On the other hand, I hear there is a plan to make our way in the world by exporting jam, so perhaps we'll be OK.