My layman summary: Silent payments allow me to post an "address" on a public website, and someone can pay me without anyone else knowing they paid me. If that's correct, this is brilliant!
Exactly. Consider the following possible use case where silent payments may be very helpful. Say, you're applying for a signature campaign or for some other paid activity where many participants are involved whom you don't trust. In the job application, you specify your silent pseudonym instead of a static bitcoin address accessible to literally everyone who can read. Other participants do exactly the same because they don't want others spying on their financial affairs. The employer makes a list of approved silent pseudonyms, imports this in his private bitcoin wallet, and generates corresponding "real" addresses when it is due time to pay. Each time his wallet makes a payment, it adds multiple unique parameters (txid, index, timestamp, etc) into the address construction process in order to prevent address reuse. If you work for ten weeks, you will end up with ten unique addresses completely unrelated to your initial pseudonym or your other addresses. Even if the employer makes his payment transactions public, it will be impossible for an outside observer to determine which coins belong to which pseudonym. In this case, even employees themselves cannot map transactions with pseudonyms due to equal amounts of some transactions. Only the employer (campaign manager) will know to whom he sent a payment, when, and how much, but he should have access to this information anyway.