Cryptocurrency provides economic incentive to support a blockchain, and a blockchain provides support to a decentralized, distributed service.
For example:
1) Money transfer - with no central point of failure or central point of control, a blockchain database provides trustless support to verify and maintain a history of transactions. But to do the work, economic incentives to workers must be applied.
Example: Bitcoin
2) Storage - once again, no central point of failure or control, with a blockchain providing the history and data regarding what is stored, and with cryptocurrency providing economic incentive for someone to provide storage to the network.
Example: Siacoin
3) Contracts - trustless contracts or escrow, maintained in a blockchain database with code for execution maintained on the blockchain as well. Verification and execution of code is done by workers who are incentivized by cryptocurrency.
Example: Ethereum
Why use cryptocurrency instead of USD? Because USD isn't electronic and is heavily regulated. As much as we like to think our money system is electronic, its really only hacky applications on top of our existing banking system. I used to work at a very large bank in the payments processing division, and you have no idea how many times they screwed up payments across the entire world. Sunday morning deployments often involved double-charges, missed charges, and overall screwing up virtually every merchant in North America. The only reason our monetary system hasn't collapsed is because people work very, very hard to fix all of these hidden failures. Cryptocurrencies remove the human element and provide a more efficient, error-free system.