Below is my original Swift 3 answer, but Swift 4 simplifies the process, eliminating the need for any casting. For example, if you are observing the Int property called bar of the foo object:
class Foo: NSObject {
    @objc dynamic var bar: Int = 42
}
class ViewController: UIViewController {
    let foo = Foo()
    var token: NSKeyValueObservation?
    override func viewDidLoad() {
        super.viewDidLoad()
        token = foo.observe(\.bar, options: [.new, .old]) { [weak self] object, change in
            if change.oldValue != change.newValue {
                self?.edit()
            }
        }
    }
    func edit() { ... }
}
Note, this closure based approach:
Gets you out of needing to implement a separate observeValue method;
 
Eliminates the need for specifying a context and checking that context; and
 
The change.newValue and change.oldValue are properly typed, eliminating the need for manual casting. If the property was an optional, you may have to safely unwrap them, but no casting is needed.
 
The only thing you need to be careful about is making sure your closure doesn't introduce a strong reference cycle (hence the use of [weak self] pattern).
My original Swift 3 answer is below.
You said:
But, in thinking about this, I don't think this will work for primitives of swift objects such as Int which (I assume) do not inherit from NSObject and unlike the Objective-C version won't be boxed into NSNumber when placed into the change dictionary.
Actually, if you look at those values, if the observed property is an Int, it does come through the dictionary as a NSNumber.
So, you can either stay in the NSObject world:
override func observeValue(forKeyPath keyPath: String?, of object: Any?, change: [NSKeyValueChangeKey : Any]?, context: UnsafeMutableRawPointer?) {
    if let newValue = change?[.newKey] as? NSObject,
        let oldValue = change?[.oldKey] as? NSObject,
        !newValue.isEqual(oldValue) {
            edit()
    }
}
Or use them as NSNumber:
override func observeValue(forKeyPath keyPath: String?, of object: Any?, change: [NSKeyValueChangeKey : Any]?, context: UnsafeMutableRawPointer?) {
    if let newValue = change?[.newKey] as? NSNumber,
        let oldValue = change?[.oldKey] as? NSNumber,
        newValue.intValue != oldValue.intValue {
            edit()
    }
}
Or, I'd if this was an Int value of some dynamic property of some Swift class, I'd go ahead and cast them as Int:
override func observeValue(forKeyPath keyPath: String?, of object: Any?, change: [NSKeyValueChangeKey : Any]?, context: UnsafeMutableRawPointer?) {
    if let newValue = change?[.newKey] as? Int, let oldValue = change?[.oldKey] as? Int, newValue != oldValue {
        edit()
    }
}
You asked:
Also, bonus question, how do I make use of the of object variable? It won't let me change the name and of course doesn't like variables with spaces in them.
The of is the external label for this parameter (used when if you were calling this method; in this case, the OS calls this for us, so we don't use this external label short of in the method signature). The object is the internal label (used within the method itself). Swift has had the capability for external and internal labels for parameters for a while, but it's only been truly embraced in the API as of Swift 3.
In terms of when you use this change parameter, you use it if you're observing the properties of more than one object, and if these objects need different handling on the KVO, e.g.:
foo.addObserver(self, forKeyPath: #keyPath(Foo.bar), options: [.new, .old], context: &observerContext)
baz.addObserver(self, forKeyPath: #keyPath(Foo.qux), options: [.new, .old], context: &observerContext)
And then:
override func observeValue(forKeyPath keyPath: String?, of object: Any?, change: [NSKeyValueChangeKey : Any]?, context: UnsafeMutableRawPointer?) {
    guard context == &observerContext else {
        super.observeValue(forKeyPath: keyPath, of: object, change: change, context: context)
        return
    }
    if (object as? Foo) == foo {
        // handle `foo` related notifications here
    }
    if (object as? Baz) == baz {
        // handle `baz` related notifications here
    }
}
As an aside, I'd generally recommend using the context, e.g., have a private var:
private var observerContext = 0
And then add the observer using that context:
foo.addObserver(self, forKeyPath: #keyPath(Foo.bar), options: [.new, .old], context: &observerContext)
And then have my observeValue make sure it was its context, and not one established by its superclass:
override func observeValue(forKeyPath keyPath: String?, of object: Any?, change: [NSKeyValueChangeKey : Any]?, context: UnsafeMutableRawPointer?) {
    guard context == &observerContext else {
        super.observeValue(forKeyPath: keyPath, of: object, change: change, context: context)
        return
    }
    if let newValue = change?[.newKey] as? Int, let oldValue = change?[.oldKey] as? Int, newValue != oldValue {
        edit()
    }
}