To your app to the globally available 'Open with...' dialogue that other apps use you need to:

1. Register your app as handling types of files (e.g. images, pdfs, etc.)
2. Implement a method that gets called in your app when another app passes a file to you

For images:

1. Register your app as accepting images by adding data to the plist file.

Add CFBundleDocumentTypes to your plist (in the Rakefile):

app.info_plist['CFBundleDocumentTypes'] = [{
    "CFBundleTypeExtensions" => ["pdf"],
    "CFBundleTypeName" => "public.pdf",
    "CFBundleTypeIconFile" => "app.icns",
    "LSItemContentTypes" => ["public.pdf"],
    "CFBundleTypeRole" => "Editor", #or Viewer
    "LSHandlerRank" => "Alternate"
},
{"CFBundleTypeExtensions" => ["png"],
"CFBundleTypeName" => "public.png",
"CFBundleTypeIconFile" => "app.icns",
"LSItemContentTypes" => ["public.png"],
"CFBundleTypeRole" => "Editor", #or Viewer
"LSHandlerRank" => "Alternate"
}]

Add UTImportedTypeDeclarations to your plist (in the Rakefile):

    app.info_plist['UTImportedTypeDeclarations'] = [{
        "UTTypeConformsTo" => ["public.data"],
        "UTTypeIdentifier" => "public.pdf",
        "UTTypeTagSpecification" => {
            "com.apple.ostype" => "PDF",
            "public.mime-type" => "application\/pdf",
            "public.filename-extension" => ["pdf"]
        }}, {
        "UTTypeConformsTo" => ["public.image"],
        "UTTypeIdentifier" => "public.png",
        "UTTypeTagSpecification" => {
            "com.apple.ostype" => "PNG",
            "public.mime-type" => "image\/png",
            "public.filename-extension" => ["png"]
        }
    }]
end

More about this step here: From Registering Your Support of File Types

2. Implement application:openURL:sourceApplication:annotation: in your app_delegate

This method is called whether or not your app is already open. (If your app isn't open, application:didFinishLaunchingWithOptions: is called first.)

This gets called when another app sends a file to your app

def application(application, openURL: url, sourceApplication: sourceApplication, annotation: annotation)
   #url.nsdata => data of the file passed
   #e.g. open a new view controller
   foo = MyImageManipulatorViewController.alloc.initWithURL(url)
   # ...
   true
 end

More about this step here: Opening Supported File Types

Note You can't add your app to the list of share options the native Photos and Camera Roll apps use.

See also

Overview of sharing/opening docs across apps:
About Document Interaction

Other document types:
CFBundleDocumentTypes