Scala has dependent type support, so, yes, it is quite more powerful than most languages in widespread use. You can look at the Shapeless library to see that at work.
However, that is not really what's going on... though Scala type system, even without dependent types, is strictly more powerful that Java's, it really comes down to the ease of use: Scala programmers tend to use types liberally (which was one of the criticisms of the article), and that is part of the "magic".
Part of this is how easy it is to define a type: it can be as short as a single line. So you'll see way less String types in Scala, for instance, or "one type to do it all" like the article wanted for HTTP. Instead, you get many types, in which invalid states are not possible and, therefore, getting the types right is enough to ensure there are no invalid states.
Likewise, the partial type inference offered by Scala makes it possible to not worry about all these types: the compiler knows what type you got, and knows whether it can be used the way you are using it, so you don't have to keep looking up types to get things accomplished, but still get to have the types.
Another part is that Scala support higher order types, something not supported by Java, which makes them more useful. That's what allows monads & companions come in and make an appearance -- it's not that you can't write a monad in Java, it's just that you can't abstract over them.
The final part is emphasis on immutability. When you stick to immutable objects representing valid states, the space for errors not caught by the type system is greatly reduced.
Let's get back to the headers, then. First, yes, you are allowed to add arbitrary headers (which is supported by Spray through RawHeader, something it took me 30 seconds to find out, even though I have never used Spray and have little familiarity with it), but section 14 of RFC 2616 has a nice list of standard headers and the rules they are expected to abide by, so let's have them.
And if you have a header, and it is a X-Forwarded-For, isn't it reasonable to expect the value to be a valid remote address? It is supposed to be a valid remote address, after all. So, if you ask for this header, should you simply get a string that may or may not be actually valid, or should the type system give you an Option that may contain Some valid header, or None, therefore ensuring that you don't just assume you are getting the right thing? And if you get something and retrieve the value, is it preferable to just get a string, that might lead to some ad hoc parsing of the address, or some actual type where you can easily retrieve information in a safe manner?
The Scala community will usually think that yes, that's a good idea, and design the APIs accordingly.