It turns out that the "Hey, Siri" feature allows you to wake your iOS device ONLY while it is charging by speaking, "Hey, Siri" and waiting for the familiar beeps. Therefore, you must charge your iOS device in order for the "Hey, Siri" command to function. Isn't it stupid that this limitation is not stated in the Settings section?
If your iPhone or iPad is not charging, hold the Home key for a couple of seconds to activate Siri.
By the way, the feature can be enabled by going to Settings > General > Siri.
Questions? Let me know!