As an example, look at "Aviation Exam". They let you buy subscriptions on-device as in-app purchases, or on their own website. In each case the details are synced to a user's account on their own server, so the same exam can be used from any device.
Look at the Apple documentation for how in-app purchase subscriptions work on iOS. Then your app can send details of a purchase to your servers, and download further information.
Edit; after discussion in comments:
If you want payment to go via Apple then it has to be via App Purchase or In-App Purchase. In-App Purchase specifically supports the idea of buying a subscription for a limited time. This is explained at the second link above.
If you want the user to create an account on your server you can either have a page in the app for them to input their details, or you can bring up a web page served from your website. Either way, the info can go to your server and it can create an account.
The key thing is, if payment went via Apple then inside the app is the only place you know this. The app can send this info to your server. You need some common identifier (i.e. a user-name) that is known to your server and to your user, then the user keys it in to your app and it can all be matched up.
There is nothing complicated here, to a decent software developer. All they need is an existence proof such as I gave at the top, and they can figure out how to link the info together.
Edit 2
Some tutorials for in-app purchase listed at: In-App purchase server model
Lots of low-level detail at: Verify receipt for in App purchase
If you prefer to handle payment yourself, not via Apple, then the situation is very different. Now, your own systems have to keep track of what has been bought, when subscriptions run out, etc. To begin with, the app won't know this at all. However, once you identify the user by having them enter credentials (username/password), you can fetch all the details from your back-end system to the app and proceed as above. Again, this is all visible in the example I gave at the beginning, which supports both Apple and non-Apple payments.
One thing to note: if you handle payment yourself then Apple isn't getting its 30% cut, which is the usual App Store commission, so they may not like this. The guidelines say:
11.1 Apps that unlock or enable additional features or functionality with mechanisms other than the App Store will be rejected
11.13 Apps that link to external mechanisms for purchases or subscriptions to be used in the App, such as a "buy" button that goes to a web site to purchase a digital book, will be rejected
That's pretty clear-cut, but since there are apps that rely on subscriptions or content purchased elsewhere, they don't seem to follow these rules in every case. Even the Amazon Kindle app was allowed back, once they took the 'buy' button off.