How does let x where x.hasSuffix("pepper") work

vegetable is an implicit String. It's the same as you would write:

var vegetable: String = "red pepper"

hasSuffix is declared as func hasSuffix(suffix: String) -> Bool an therefore returns a Bool. The where keyword specifies additional requirements, and can only be used in switch statements.
Because all of this is suffused, the vegetable variable is assigned to x (let x).

You can read more about the where and switch here.