I don't quite understand how the server can match the ciphertext with a value without knowing the key. How does the server determine that the ciphertext corresponds to the specific value? If the server constructs this ciphertext-value database, how does it know what algorithm to use to create ciphertext from a value and store on its side?

Check my comment here for some intuition: https://news.ycombinator.com/item?id=41115179

Basically the server does not know, it just computes with every possible value. And the result turns out to be what the client was interested in.