You and me are sitting on a bench in a park, we decided to exchange 100USD, 5 other friends present in the park works as a proof that we have made the exchange.
Cryptocurrencies eliminate the need of a centralised 3rd party(banks) for transferring money.
Cryptocurrencies run on Blockchain, which is a decentralised ledger where many people who are part of the network makes sure that a transaction is done between me and my friend.
Hope I have made it clear to you.
Also if you are interested in buying any crptos, you can have a look at CoinSwitch. They provide you with the best possible rates to trading.