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On the Ethereum Network, ETH transactions are authorised by digital signatures linked to public key addresses. When a user Bob wants to send ETH to Alice, he initiates a transaction by digitally signing it with the private key corresponding to his public key address. This signature proves ownership of the ETH, and can be easily verified by other members of the Ethereum network.
Once broadcast to the Ethereum network, Bob’s transaction is verified by network participants. They confirm Bob's ownership by checking their local database (a trie structure) to ensure he has enough ETH to cover the transfer. If he does, and his transaction is otherwise valid according to Ethereum’s protocol rules, Bob’s transaction will be relayed by each network participant, making its way throughout the Ethereum network.
It is the digital signature(s) alone, created by his private key(s), that proves to the network that Bob owns the coins being transferred. This way, transactions can be easily and cheaply validated by anyone on the Ethereum Network without having to involve real identities.