How exactly does "silent" authentication using matching local accounts on a Windows network work?

It's called pass-through NTLM authentication and is documented in the NTLM TechNet article. See the section titled Pass-through authentication toward the bottom.

The NetLogon service implements pass-through authentication. It performs the following functions:

• Selects the domain to pass the authentication request to.

• Selects the server within the domain.

• Passes the authentication request through to the selected server.

Selecting the domain is straightforward. The domain name is passed to LsaLogonUser. The domain name is processed as follows:

• If the domain name matches the name of the SAM database, the authentication is processed on that computer. On a Windows workstation that is a member of a domain, the name of the SAM database is considered to be the name of the computer. On an Active Directory domain controller, the name of the account database is the name of the domain. On a computer that is not a member of a domain, all logons process requests locally.

• If the specified domain name is trusted by this domain, the authentication request is passed through to the trusted domain. On Active Directory domain controllers, the list of trusted domains is easily available. On a member of a Windows domain, the request is always passed through to the primary domain of the workstation, letting the primary domain determine whether the specified domain is trusted.

• If the domain name specified is not trusted by the domain, the authentication request is processed on the computer being connected to as if the domain name specified were that domain name. NetLogon does not differentiate between a nonexistent domain, an untrusted domain, and an incorrectly typed domain name.