Many of you may already know this but I'm going to write this up for those who may not because there seems to be a lot of confusion.
The cloud is a term generally used to mean centralized remote data center. In the prior methodology you might have a physical server or set of servers that run your server software then a set of clients that connect to it. These servers were typically local and you needed to have powerful enough hardware to run at peak utilization, probably with a little room to grow. The problem here is that often load isn't even and there are peak times and then times with minimal usage (such as overnight). So you have this big expensive set of hardware sitting around doing nothing over night for example, that isn't cost effective.
The cloud resolves this by having a data center with a lot of hardware were the hardware is shared between multiple virtual servers. In a simple example lets say I have a big server with 16 cores and 1 terrabyte of RAM. I can then put virtual machines for servers on this physical machine where each server gets 2 cores and 128GB of RAM. I only have one physical machine but to the outside world it looks like I have 8 servers. It gets even better than that though because you don't have to be so exclusive. Odds are those 8 virtual servers don't use that hardware to its full potential 24/7 so if you're smart you can put even more virtual servers on that hardware whose peak usage times line up with the low periods of the other servers. This way you can maximize the usage of this expensive hardware you have. Now if you add to that a duplicate system for load balancing lets say. So you have two physical machines that appear identical to the outside world (including all the virtual servers running on it) but the load is balanced between them. This means when you start getting near the limits of the hardware you can shift everything over to one server while you beef up the hardware of the other then do the opposite so the virtual servers never go down but now they are running on even more powerful hardware. This either means your existing virtual servers have more power to work with or you can add even more virtual servers (or both).
This is FAR more cost effective and in fact the providers of these cloud services may only bill you for what you actually use. All the technical stuff is abstracted so instead you just pay for processing time, storage space, etc. Now you can pay a little more during peak time for better hardware (such as at launch for a new game) without having to BUY super expensive hardware to handle it and then as usage drops down your costs likewise fall and you aren't stuck with a bunch of hardware you don't need. The cloud provider just resells the capacity you aren't using to someone else and everyone wins.
Clouds are a big thing in business right now. My work for example is getting ready to require all our servers worldwide use a private cloud they are setting up. So our server admins will still be local but they won't work with hardware anymore. We'll "buy" cloud capacity from this common cloud where they will set up a virtual machine for the servers that our admins will administrate locally. The physical servers are in another state, possibly another country, but our admins our still local and the hardware is handled as a service by the "cloud" providers.
MS is absolutely right that this makes a ton of sense of game servers as well. What is confusing though is that MS makes is sound like it's something new to them or specific to the Xbox One. Nothing stops a game developer from setting up their own cloud or buying capacity on an existing cloud service (such as Amazon's or Microsoft's own Azure) and using that to run servers for PC, Mac, Linux, PS4, PS3, etc. You'll note I didn't say Xbox there because Microsoft prevents developers from hosting servers outside of their Xbox Live network so they can NOT use existing cloud services. To resolve this MS created an Xbox Live cloud (confusingly still called Azure because they're based on the same basic technology). So MS has conceptually two different Azure clouds, one on the public internet that we'll call the Commercial Azure cloud and another on their Xbox Live network that we'll call the Xbox Azure cloud. The Xbox Azure cloud is usable only by Xbox One games and has some game specific extensions as a result. The Commercial Azure cloud lacks game specific extensions but is usable by anyone who wants to buy capacity from MS. MS isn't going to refuse to sell you capacity on their Commercial Azure cloud because you happen to be say a PS4 game developer so again nothing stops other consoles from using cloud services for servers. You do lose the game specific extensions but they're like Xbox specific anyway so if you're making a cross platform game you're going to have to avoid them anyway for your non-Xbox clients.
So the Xbox Azure cloud is a really cool thing for Xbox Ones because it allows them to finally have dedicated server-like capability at a cost effective rate. However if you use it you're tied to Xbox only (MS may allow Windows PC's to connect to the Xbox Live cloud as was the idea for the now-defunct Games for Windows Live service but I'm not sure about that). Also the Xbox Azure cloud is your only cloud option as a Xbox One developer so thank goodness MS's Azure technology is REALLY good. Whose to say 5 years from now someone else won't crush them though? On the other hand a developer could set up their servers on any public cloud service (including MS's Commercial Azure cloud) and have it usable by Windows, Mac, Linux, PS4, and even PS3 users to name a few (pretty much any platform that connects to the public internet). For bigger companies it may even make sense for them to set up their own Clouds. This is especially true with big publishers like EA. EA could set up their own data centers for the games they publish and it could host servers for all their various multiplayer games. The same set of hardware could provide virtual servers for multiple games and multiple platforms (except Xbox which MS won't allow to connect to public servers... just Xbox Azure). So PC, Mac, PS3, PS4, etc. EA gamers could be running on a common cloud platform with the Xbox ones would be running on the Xbox Azure.
Cloud computing IS the future, MS it correct on that. It is the future however of network computing in general both business and games and is not in any way specific to Xbox One. Offloading AI, etc. to the cloud is no different than a dedicated server doing the AI for NPCs that's existed pretty much since multiplayer gaming began. The cloud just makes dedicated server-like capabilities more cost effective.