Why is my iPhone screen stuck on the 'connect to iTunes' screen?

Please note that the instructions I am giving are assuming you are using an English copy of Windows, since I do not have a Mac and therefore do not have much experience with them. However, the steps will be very similar, and I will try to note where things might be different for Mac OS X.

First, check all of the following:

• Make sure your copy of iTunes is up-to-date. In Windows, open iTunes, click "Help", then "Check for Updates".
• Make sure you are connected to the internet and the connection is reliable. iTunes contacts Apple several times during the restore procedure, and if iTunes fails to get a response during one of these check-ins, the restore will fail.
• Make sure you are using the correct version of iTunes for your system. If you are using a 64-bit version of Windows, you will need to install the 64-bit version of iTunes. If you have Windows XP, you need the normal 32-bit version. If you have Vista or Windows 7, click the start orb, right-click computer, then click "Properties". In the "System" section, look for the line that starts with "System Type". If it says 32-bit, use the normal iTunes installer. If it says "64-bit Operating System", download the 64-bit version of iTunes from http://support.apple.com/kb/DL1047 (If you have trouble determining whether you have a 32-bit or 64-bit copy of Windows installed, try installing iTunes using http://www.ninite.com and it will automatically download the right version)
• Delete the restore files from your computer. In Windows, press WindowsKey+R, type '%appdata%\apple computer\itunes' then delete the folders iPod Software Update and iPhone Software update if they exist. If you are using OS X, the folder should be ~/Library/iTunes/.
• If it still isn't working, uninstall iTunes, reboot, then install iTunes again, then reboot again.

If none of the above work, do the following to place your iPhone into DFU mode. Some people have better luck with restoring with this method.

1. Plug the iPhone into the computer, and wait for it to be recognized in iTunes.
2. Press and hold the home and sleep buttons.
3. As soon as the iPhone screen goes blank, count to six seconds. At six, release the sleep button, and continue holding the home button.
4. You should see or hear the new hardware notification on your computer. When you do, you can release the home button. If the new hardware notification, wait until iTunes recognizes it and tells you it found an iPhone in restore mode.
5. The iPhone screen should be blank. If it shows the "Connect to iTunes" screen, unplug it, then try again from step 1.

If you see Connect to iTunes screen when you start iPhone, iPad, or iPod touch

In some situations, you might see the Connect to iTunes screen when you start your iPhone, iPad, or iPod touch. When this happens, your device is in recovery mode. Here's what to do.

Restart your device by pressing and holding the Sleep/Wake and Home buttons at the same time until you see the Apple logo.

If you see the Connect to iTunes screen again, you'll need to reinstall iOS. Follow these steps: 1. Connect your device to your computer. 2. You should see this message: "The software on [your device name] needs to be restored to factory settings or updated." 3. Click Update to reinstall iOS while keeping your data.

Get help if, after following the steps above, you see a restore error.

If you don't see a restore error, but you still see the Connect to iTunes screen when you start your device, contact Apple.

Source - https://support.apple.com/en-gb/HT203122