"What is Bitcoin backed by?"

It is backed by value.
Value is created by people, it is not intrinsic to anything except ideas (opinions).
Certain resources (gold, dollars, real estate, stocks, etc) have great monetary value in our society because we look for, or create, certain properties in them when following a particular - the most popular - economic theory at the time.
As an example, if we all started to adopt the theory of resource-based economics, widely promoted by Peter Joseph who created the Zeitgeist movie-series, we would value resources very differently.
So what makes Bitcoin valuable to people (adopters) in a capitalistic economy?
It can be a multitude of reasons:

• Deflation is better than inflation
• It is better to have a decentralized power
• Transactions in Bitcoin are cheaper, easier in a digital world
• The Blockchain: a PUBLIC record of every transaction
• I hate Ben Bernanke or the Federal Reserve System
• The security of Bitcoin's cryptography
• Irreversible transactions
• A new economy to start a bushiness in
• Anonymity (if done properly)
• Dude, I can buy drugs like I buy shit on Amazon... awesome
• I don't know, my friend uses it and I just follow whatever they does cause I think they're cool (majority of people)
• I like code
• I like turtle
• Miner: "Free money man!"
• we can win a major battle in the arms race and gain a new territory of freedom for several years
• Etc...