From the iOS guidelines, any app which downloads code in any form is (or at least should be) automatically rejected for the App Store. Most likely is that your app is downloading something which would be covered under the concept of 'content.' Depending on the particular app, this could be new images, sound effects/music, new structural descriptions (ie a file describing a new zone in your game, specifying how the game constructs it from existing resources, or resources included as part of the update), or any number of similar items. Basically, the AppStore component of the program will contain the logic used to construct and control the app, while the data being downloaded controls the specific content being shown at any given point in time.
From your example (Game of War), being an online multi-player game, it would likely contain code in the AppStore component which controls the interaction between the server and your device, how to send and receive chat messages, player actions, etc. I haven't played this particular game myself, but it appears to be using the standard gridded map model - each building (including it's phases of construction), and each unit on the map could be generated by the AppStore component based on downloaded image files and animation descriptions. Your game can incorporate updates including new unit types, or new buildings without having to change any code on the device, simply by downloading the components necessary to display them.
The structure is similar to how many MMO's (like World of Warcraft, etc) work on the PC, although the fact that interpreted code is still a fairly grey area in regards its legality in iOS apps (see this question) means that the apps are less likely to be downloading interpreted components which can change the fundamental interactions with the app (ie the actions you are able to perform in a game, or including a chat feature into an existing game, unless it was designed for and just not 'activated' in the initial release) and more just including new items which match one of the existing patterns of understood elements.
You will probably find this type of interaction more in games, where the coded logic makes up only a small fraction of the total size of the game and the majority of the size is the images, sounds, 3D models, structural descriptions, etc which the game uses to construct its internal environment, than in business apps. Essentially though, the inbuilt Maps and Music apps follow a similar concept:
- your audio tracks would be the equivalent of a downloaded 'update' in Music
- the idea of downloading purchased content to a new device through iTunes
operates on the same principal
- while I believe Maps on the iOS downloads the map images directly
from a server, similar software in portable GPS devices use an
internal store of map images and road descriptions which can be
updated in this fashion to provide up-to-date information on new
roads without having to change any compiled code
- even the AppStore itself works under the same underlying theory, if you
consider that the other installed applications are essentially it's 'content'
(the AppStore can download an 'update' to another installed app without having
to change any of its own code)