iPhone 5s blue screen,

First, try to reset the phone by holding the sleep and home button for about 10sec, until the Apple logo comes back again. You will not lose data by resetting.

If this does not help, set it up as new device, explained here:

Use iTunes to restore your iOS device to factory settings - Apple Support

If still no luck, get it checked by visiting an Authorized Apple Service Provider or contact Apple Support to get it serviced:

iPhone - Contact Support - Apple Support

I had the same issue with my iPhone 5S. After calling AppleCare and going to 3 different genius appointments, one of the Genius said he knew the problem. He said it's a problem introduced after iOS 8.x where the firmware is problematic. All my updates were OTA (aka patches), and so the firmware wasn't correctly patched or something. Anyway, even though my phone was stuck in recovery mode and claimed by other Genius' that a $269 replacement was the only way, this one Genius was able to get the phone into DFU (only way to replace the underly firmware. Recovery mode does not replace problematic firmware) and completely replace the erroneous firmware. Phone works again!!!! Hope this helps you guys!!!!