This question is often asked and the answer varies depending on who is being asked, so, what DOES back cryptocurrencies, what underpins them, where is their value coming from. 

First to the stock responses: 
Cryptocurrencies are backed by the technology and mathematics behind them, they are backed by the confidence reposed in this technology, but this is not enough; to truly answer this question we have to first explore other forms of money (cryptocurrencies are also forms of money).

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So, what is money, what underpins it and where does it get its value from? We shall examine these questions relating the arguments and reasoning that we come up with to cryptocurrencies to see  if and how they apply and to what measure.

What is money - there are certain core functions that money has to be able to fulfil before it to be considered money,  it has to be a store of value, it has to be transportable, i.e. it has to be easily moveable either in physical or digital form, it has to be a very good means of account, it also has to be divisible – money has to have these core functions in order to be considered money - cryptocurrencies share these functions with money in equal measure.

Money is underpinned by the economy of the nation that it belongs to but what really gives it its value is the manipulation that can be applied by a central body, i.e. central banks. 
The value of money is heavily influenced by the actions of central banks, we saw this demonstrated with “quantitative easing” in the late 2000s when money was just created out of thin air to shore up the American economy.

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So, what DOES! underpin cryptocurrencies, what underpins cryptocurrencies is trust, trust in the mathematics, the science, the algorithm, the technology, trust is in the rules that guide the market place and the parameters that has been set around it.

Most cryptocurrencies have set rules on how they can be created (e.g. mainly through mining), in some instances how many would ever be created (Bitcoin has an upper cap of 21 million), and because virtually all cryptocurrencies are  decentralised a global community of people are able to monitor them and ensure that the rules are adhered to. 

Cryptocurrencies are backed by virtually unbreakable encryption codes, this serves to reassure people that it is safe and secure, it would cost more to fraudulently carry out a transaction in cryptocurrencies than the fraudulent transaction would be worth, I have heard estimates of close to $1 billion as the cost of being able to code break a currency like  Bitcoin.

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 Cryptocurrencies are the most transparent form of money there is, this transparency is a consideration / factor in the trust people have in them, people can see the inner workings of the currencies on open source blockchains this serves to reassure and helps build confidence in the market place.

Admittedly the market place is still very volatile  at the moment because it is still at its fledgling stage however, as it continues to grow it will quickly stabilise. It is clearly set to be the currency of the future. 

In summary the most important thing that underpins / backs cryptocurrencies is the free market, it is backed by the people who have chosen to confer trust and value on it, the fact that it is a decentralised market place is a huge part of this, the fact that it is not centrally controlled and it is a “democratic money” globally owned by people who believe in it is what backs it.