Your iPhone camera's shutter sound effect is controlled using the same volume controls as your phone. This means that you cannot mute the camera sounds without also muting the volume for the rest of your phone. You can mute and unmute whenever necessary by using your iPhone's Ring/Silent switch, but you can also use workarounds like plugging in headphones or using your Music player.
Close your camera app and press your iPhone's Home button to view your main screen.
Locate the Ring/Silent switch right above the volume control buttons on the side of your iPhone and switch it off. You will know it is in the mute position if you can see an orange bar on the switch. You will also see an icon appear briefly on your phone's screen showing a bell with a line through it.
Open the Camera app and take pictures normally. All sounds from your phone should be muted, including the camera's shutter sound.