IN PLAIN ENGLISH:
The only reason any currency, paper or digital, has any value is because people agree that it does. How much value it has simply comes down to how many people want it versus how much of it there is to go around (supply and demand).
The technical explanation:
People are used to the idea that paper (fiat) currency has real value as money, but isn't it just a piece of paper in the end? Why is a $100 bill worth twenty times more than a $5 bill when it's made of the same thing? Simply because we all agree that it's worth that. Someone will gladly give you twenty pieces of paper they believe have a value of $5 each in exchange for one piece of paper they believe is worth $100.
Cryptocurrencies operate the same way. At the time of publication, one Bitcoin is worth roughly $14000 USD. That is currently how much people are willing to pay for one, so that's how much it is worth. In the case of Bitcoin, which is by far the most popular cryptocurrency at present, there is a maximum of 21,000,000 that will ever exist. This makes it scarce because there is a limited supply. As Bitcoin becomes more mainstream, and widely accepted, the demand for it will continue to increase while the supply stays the same. This limited supply versus the increasing demand is what causes it's value to rise.
This is the same idea as gold or diamonds. Because these are highly desired items, and they are scarce, the value of them is very high. If it was suddenly possible to mass-produce diamonds in a factory causing there to be a huge supply of them, they suddenly wouldn't be worth very much. Anybody would be able to get one at a very low price because there would be more supply than there is demand.
The difference between fiat currency and cryptocurrency is that fiat currency is created and controlled by the federal government of countries. With fiat currency, banks act as a trusted third party to safely store your money, keep a record of who owns what, and make financial transactions. With cryptocurrency, no bank or government is needed. The blockchain is designed to create the currency automatically at predetermined intervals through mining (up to the set limit), which is actually more reliable than governments who can just print new money whenever they decide to, which diminishes the value of the currency. The blockchain uses cryptography to automatically verify transactions and transfer funds, and the Distributed Ledger keeps track of who owns what--all anonymously and without need for a trusted third party. Trust is built into the system due to the brilliant technology that is the blockchain.
Cryptocurrencies are being used as an investment because it is expected that the value of cryptocurrencies will continue to rise as more of the general public learns about it and wants to take part. This is an exciting time and space to be involved in! Anything is possible, and things are changing rapidly.