How do I use variables in a localized string in Swift? 

Interpolate the variable inside String(localized:), available since iOS 15 and macOS 12, or inside a SwiftUI Text literal. The key Xcode extracts contains a format specifier in place of the value, and the translation keeps that specifier.

let name = "Ana"
let count = 3

let greeting = String(localized: "Hello, \(name)!")
let summary = String(localized: "You have \(count) new messages")

When you build, Xcode adds Hello, %@! and You have %lld new messages to the String Catalog. A String becomes %@, an Int becomes %lld and a Double becomes %lf. The translator writes the words around the placeholder: "¡Hola, %@!". The initializer is documented at String(localized:table:bundle:locale:comment:).

SwiftUI needs no extra call, because a string literal passed to Text is a LocalizedStringKey:

Text("Hello, \(name)!")   // looks up the key "Hello, %@!"

Before iOS 15, use NSLocalizedString for the format and substitute the values afterwards. Never interpolate inside the NSLocalizedString key: Xcode cannot extract it, and translators receive the code expression instead of the text.

let format = NSLocalizedString("Hello, %@!", comment: "Greeting with the user's name")
let greeting = String.localizedStringWithFormat(format, name)

String.localizedStringWithFormat formats numbers for the user's locale; String(format:) substitutes them without localizing.

Pitfalls:

• With two or more variables, a translator may need to change their order. Positional specifiers make that possible: the translation "%2$lld messages from %1$@" takes the same arguments as the source "%@ sent %lld messages". The numbers refer to the interpolations in source order.
• The type in the specifier must stay the same in every translation; a %@ replaced by %d can crash the app.
• String(localized:) searches the main bundle by default; pass bundle: for strings that live in a framework or package.
• A count that changes the sentence between singular and plural needs a plural rule, not two strings; see How do I localize plurals in an iOS app?.