So let's look at the transaction chain for fiat currency in general. For the connected world (i.e., any non-cash transaction), the ownership looks like this:
[Your funds] => [stored in an online account register with a bank/financial institution]
You do this for 1) security (since banks/FIs have insured accounts), and 2) convenience.
What most blockchain currencies do is automate the security of your online account by providing you with a personal encrypted register of that account. That solves #1. But what about #2? That's where ethereum (and some select other currencies) come in. A transaction model with a traditional bank looks like:
[buyer's money] => [Payment processer (debit, credit, ACH, wire transfer, etc.] => [payment processer] => [escrow/settlement/etc. (for all but the wire transfer)] => [seller's account]
Now, here's what that transaction looks like using a smart contract with a cryptocurrency:
[Your wallet] => [seller's wallet]
There are fewer intermediaries, fewer steps to get funding, the transaction is fully public and easily auditable, and more direct flow of funds. It's a pretty simple concept; right now, banks are intermediaries that stand between buyer and seller, and take a processing fee as payment for the convenience of allowing you to spend your money.
With an autonomous cryptocurrency as the primary store of funds, there would be no need for banks, and as a result, fewer intermediaries in transactions, and fewer transaction fees, with comparable (if not better) security, using a simplified, easily auditable, and straightforward payment scheme.
Banks are already investing in this concept as an alternative to payment processors like Visa, MC, etc. It's already being implemented in the "pay a person" tools banks offer through VenMo, ClearXchange, etc. The logical next step is to simplify that process by removing banks/fiat currency.