Similar to taking protein after a workout, drinking your shake before bed is based on the idea that your body repairs muscle more effectively during sleep. Of course, protein is required to repair muscle, and so it makes sense to supplement just before rest.
We would also suggest looking at casein protein powder as an ideal pre-sleep option. Its peptide and micelle structure is disturbed by the acid in your stomach and this causes the casein to form a gelatinous structure which slows the digestion process. This means that rather than flooding your body with essential amino acids, it forms a steady flow of amino acids over a far longer period of up to 7 hours. While this may not be useful after a workout, it makes casein ideal for its anti-catabolic effect while sleeping.
There are a couple of question marks over this idea, the first being how well you might sleep when you’re digesting - as is required before you can start repairs. The second issue is how much your bladder can hold, and if the answer is “not much”, then you’d best steer clear of this late-night snack unless you’re up for a midnight trip to the bathroom!