How can I silence the screenshot camera sound?
To my knowledge, there technically is not a way to specifically and only disable the camera shutter sound for screenshots, but there are still solutions to stop it from making the sound:

One thing you can do is flip the mute toggle switch on the side of your device to be muted (silent/vibrate). You will see a Ringer Silent icon on the screen and may feel a vibration on an iPhone. After that, there will be no noise from the screenshot (among other things).

On an iPhone, if you don't wan't to get the vibration when turning the phone to silent, or if you want to leave it on silent without getting vibrations, you can go to Settings > Sound and toggle "Vibrate on Silent" off. Then it won't vibrate or make any noise. However, you also won't hear any notification sounds or vibrations at all, so you run the risk of missing some alerts. So in this case what you may want to do is flip the mute toggle to silent right before taking a screenshot (and it won't make any noise from vibration), and then after the screenshot flip it back to normal.

Another alternative, if you don't mind the camera shutter sound but just want it quieter, is that in Settings > Sounds you can slide Ringer and Alerts to minimum. It'll still make sounds, just quietly. But again you run the risk of missing your phone ringing in some situations as it'll be quiet.

Edit: I noticed that on iOS 10.3, when you toggle the mute switch while the camera is open, it does not make any vibration. So you could just toggle it to mute before you take the picture, and then toggle back to normal when you're done.