It could be anything from the physical appearance on yourself/others to the mindset of losing control when wearing them/ gaining control over others when applying them, or anything else you can think of. I find myself curious how other braces enthusiasts think.

I always think I'm a bit unusual here in that I have no desire to wear braces myself, but like to see others in them.
As to what I like about them, the look does have something about it for sure, but I like the inherent awkwardness that goes with wearing braces, and the loss of personal control in what happens to you.
A particular favourite 'scene' of mine is when a very confident guy suddenly find himself with a greater than average amount of metalwork, and how he changes as a result.

There’s a lot going on.  The “control” is definitely a thing, especially on something like headgear.  With straight braces, one gets used to them pretty quickly.  But wearing a headgear takes some effort, and nobody ever gets “used to it”.  The discipline needed to keep that thing strapped to your face takes some doing.  Having that control imposed and/or imposing that control on somebody else is quite a trip!