In order to deploy code (without jailbreaking), you need to have it signed by a certificate that is valid for the device you are using. With a normal developer account, you can provision certificates for up to 100 devices. Signing an application is fairly simple (I implement most of the intricate pieces in an open-source program called ldid, but do not do the official signature parts).
Putting this together: you simply get paid developer accounts from Apple (you will need to keep getting more of these, as you will run up against the 100 device limit), ignore the part of the contract you sign that states that you are not allowed to redistribute the certificates they give you, and instead automate a process to log in to Apple's portal, add a UDID to your account, and download an updated provisioning certificate.
Connect this up with a payment processor that is willing to look the other way and an app that is capable of doing the signature process for apps locally for the user, and you are done. The device is already capable of installing certificates and applications that are signed by them from websites (see Test Flight for a commonly-used legitimate example), so the final steps are easy.
(With an Enterprise account, you don't have to worry about the number of devices; however, the contract terms are much stricter, the system has more mechanisms for Apple to turn you off, and Apple will probably care more if you attempt to abuse it. In essence, an app signed by an Enterprise certificate can be installed on any device, anywhere, at any time until the certificate's three-year expiration date hits.)
Given that developer accounts cost money, and paying requires giving personally identifiable information, what keeps Apple from seeing that you are buying multiple accounts?
So, first off: I really do not believe that this service (Zeusmos) was that popular; I mean, when you hear about the number of people who were using Installous (omg too many ;P), the number of people using Zeusmos would be this tiny tiny speck in comparison.
Once you are only talking about hundreds or even in the low thousands of devices (as opposed to tens of millions), you can satisfy the demand by getting a bunch of friends from your high-school class (the developer of Zeusmos was 15) to register for individual developer accounts and then contribute their Apple ID and passwords to a pool.
Based on what it says in the article (edit: which a friend of mine is telling me might be wrong, so maybe these people are doing something more complex), the other product (Kuaiyong) was using a single Apple Enterprise certificate: you only need one of these to satisfy an infinite number of devices, and they probably were signing the stuff on their server rather than on the client (so not redistributing the certificate).
That said, the rules on how you can use an enterprise distribution certificate are quite strict: you can sign applications used only by 1) employees of your company; 2) customers of your company who are physically present at your company's place of business; or 3) customers of your company who are being physically supervised by an employee of your company while at another location.
Putting this together: you simply get paid developer accounts from Apple (you will need to keep getting more of these, as you will run up against the 100 device limit), ignore the part of the contract you sign that states that you are not allowed to redistribute the certificates they give you, and instead automate a process to log in to Apple's portal, add a UDID to your account, and download an updated provisioning certificate.
Connect this up with a payment processor that is willing to look the other way and an app that is capable of doing the signature process for apps locally for the user, and you are done. The device is already capable of installing certificates and applications that are signed by them from websites (see Test Flight for a commonly-used legitimate example), so the final steps are easy.
(With an Enterprise account, you don't have to worry about the number of devices; however, the contract terms are much stricter, the system has more mechanisms for Apple to turn you off, and Apple will probably care more if you attempt to abuse it. In essence, an app signed by an Enterprise certificate can be installed on any device, anywhere, at any time until the certificate's three-year expiration date hits.)