Many iPhone 7 users complained of activation problems when the phone was first launched. The problem has since reduced and fewer users may come across an iPhone that does not activate in the first go, but if you are facing a problem like this then you can visit https://www.apple.com/ support/systemstatus/ to check the status of iOS Device Activation. If you see a green button next to it then the system is active and you should be able to activate your device.
Solution 1: Switch your internet connection
If your phone does not activate even though the system status on Apple’s website shows a green light next to iOS device activation then you must try to switch your internet connection. If you were trying to activate you phone over Wi-Fi then switch to mobile data and check if it works. Similarly, if you were already using cellular data then try connecting to Wi-Fi. There is a chance that your current internet connection may be weak.
Solution 2: Reboot your phone
Press and hold the Sleep/Wake button and then drag the slider so that your iPhone 7 switches off. Wait for about 10 seconds and then restart your device by pressing and holding the Sleep/Wake button.
Solution 3: Activate your phone through iTunes
Update iTunes to its latest version on your computer and then open iTunes. Plug in your device and once iTunes recognizes your phones, go through the activation procedure through iTunes.
If nothing works then you must consider taking your phone to the Apple store where they can check for deeper problems or any hardware issues that may be causing the issue.