iOS App Store: Unable to download and update apps due to Terms and Conditions loop, even after agreeing
Important Update

As noted in the below linked 9to5Mac article, an updated has been shared by TechCrunch:


  Update: TechCrunch has now reported that Apple has “deployed a fix” and the bug should be resolved “in a matter of hours” as the patch is rolled out. The fix is being applied server-side, so users won’t need to make any changes to see the problem resolved.




Originally Posted Answer

This is apparently an issue currently with the App Store which is affecting a large number of users and is also affecting Apple's own servers. It is widely reported.


Some users unable to download or update App Store apps, faced with endless iTunes terms and conditions prompts - 9to5Mac



  Some sizeable number of people are experiencing an issue with the App Store at the moment. On an affected account, when attempting to download a pending app update or a whole new app, users are presented with a dialog to accept new Apple Media Services Terms &amp; Conditions.


and


App Store Users Report Endless T&amp;C Loop Preventing App Updates and Downloads - MacRumors



  A number of iPhone and iPad users this morning are taking to Twitter and Reddit to report an issue with the App Store that prevents them from downloading or updating apps. 
  
  The problem starts when a user taps the Get button on an app listing, whereupon a pop-up informs them that "Apple Media Services Terms and Conditions have changed" and that they must read and accept them to continue.


According to the article, some users were able to circumvent the issue as follows:


  Some users report that simply tapping Cancel when the T&amp;C notification first pops up makes it go away, allowing them to download or update the app in question.


Since it's a widely reported issue, it's very likely that a fix will be made for the same. You can try the aforementioned, or wait for some time to have the fix available.

You can also consider getting in touch with Apple support via their support channels:


Apple Support (@AppleSupport)| Twitter
Contact - Official Apple Support


Do note that contacting support is crucial here. Even if the contact personnel is not able to provide you a solution instantly, with enough number of requests going to Apple Support, they become aware of the gravity of issue and will work on fixing it soon.