Earlier quoted context omitted.
A second layer transaction does not involve the Bitcoin network. When Katy wants to pay ₿3 to Billie, Katy just sends a key (via email or some API) to Billie which enables Billie to withdraw ₿3 from the account of Katy. But Billie does not do that. Billie just keeps that message like you keep your login to your bank account. When Katy wants to pay another ₿5 to Billie, Katy sends a key to Billie which enables Billie…
However, if nothing happens on the Bitcoin network, then the funds aren't locked down, are they? In other words, Billie has no guarantee that the key she holds will be valid tomorrow; Katy could empty that account at any time, and Billie would then hold nothing except the evidence of Katy's wrongdoing.
they are locked down. the funds in a lightning channel are locked in a 2 of 2 multisig address, and can only be moved upon mutual consent (ie. by updating the state of the channel), or by one party after sufficient time has passed (in case of an uncooperative peer).