Imagine you want to pay someone privately. The most common way to do this would be for them to generate a new address and send it over some encrypted channel. But what if this isn't possible/available?

Or put another way, what if I share my address here, in a public setting (bc1p57qdmjghng5xjugna0pjdy7ee9umce9x4jmwl0wn97804et489rq2j3f6t). Is it possible for you to pay me without anyone being able to find out that you paid me? With Silent Payments, this is infact possible!

Silent Payments are a new form of stealth payments (there have been proposals before, but some of them were not as efficient because they required OP_RETURN outputs to share a shared secret), or required a full node to be running. This new proposal is much simpler, and can look indistinguishable from normal transactions, and relies on fairly simple math. I've actually tried simplifying the math in a previous comment: #191656, hopefully it makes sense