I'm not sure what exactly is it with braces but I think it's mostly the feeling and the awkwardness .

It's about not being in control. No matter what personality you have, once you have braces you have to sit at the orthodontist with your mouth open and have no choice but to accept what someone else is doing to your mouth. Not knowing what is going to happen in an appointment. You might end up with some sort of elastics, sometimes a thicker wire or other stuff. But many times you will only find out while you are sitting there with your mouth open.

Ypu described my experience so well. The length of my treatment only came up after the fixed braces were applied and wired. The headgear I received was never discussed and simply installed. I remember stepping out of the office into a full waiting room with my headgear, just waiting for the ground to open up and swallow me. It didnt, instead it created vivid memories and probably the reason I'm on this forum. 

For me, I think, it's the slight imperfection in the beauty of a woman (I like people who have braces, but don't want to wear them by myself). I also like crooked or slightly stained teeth.

Maybe it's easier to understand using another topic as an example: modern/contemporary classical music is often dissonant, but very interesting to listen to. Perfect aesthetic "too much" beauty tends to be boring, divergences give each one of us his_her individual charme. Excuse my bad englisch, I'm not a native^^

I think it’s the control aspect for me. I don’t really like the idea of having braces but it’s something medically necessary that is controlled by someone else who is basically in charge of your comfort.

I have autism and that could be why I love hearing aids, dentures, retainers and eyeglasses with very thick lenses?   I am 29 and have my own backyard playground with 5 slides and a 12.5 foot tall commercial grade swingset.  . 

I also convinced my wife to get braces which was great. She doesn’t feel the same way about braces as I do which is unfortunate.

Its a very difficult question to answer, although I have tried to explain it at various times in other posts I suspect.
Firstly I like removables, basic Hawleys etc really.
It started at an early age maybe around 10, possibly earlier. I do remember my grandfather and then my father getting dentures and somehow I wondered how that would feel.
I remember someone in my class at primary school getting a 'brace' (retainer as I know it now).
I moved to a different town aged about 11 and was given a desk to sit at next to a boy who wore a retainer. I dont remember much about that except he would take it out for pe lessons etc.
I remember thinking it would be interesting to wear something like that and see what it felt like. Especially what it would be like not to be able to feel the roof of my mouth and what it would be like to talk or eat.
Others began to get similar 'braces' (retainers). I remember one who took two out of his mouth but at that time I dont think I realised there were top and bottom retainers available. Some had brackets and wires, ie. fixed braces, but apart from slight interest they didnt really hold any appeal to me.

I discovered that a closer friend had to wear a retainer at night. He came round to our house wearing it one day. I wanted to ask about it but didnt.

I think I used to experiment by chewing up toffee or other chewy sweets (candy) and pushing it in to the roof of my mouth to see what it was like.

And so the interest went on. Always it was a case of seeing what it was like to have something different. I willed my parents to decide that I should have a 'brace' (retainer) and my teeth would have benefitted as they were not perfect, but no one ever really thought of it and I could hardly suggest it!

Then there was a long gap until I discovered the internet and I found a company to make me what I wanted. I acquired one, then a few others. I wore one plate, which didnt show, for a number of years full time and the others when I chose...and still do. I enjoy the 'different' sensation and the fact that it is something to 'get accustomed to and deal with' but unlike many people here, its not really a case of letting them show to others or even much about others being forced to wear them. It is something about learning to adapt to them.
A lisp is something to be overcome and, in fact, I really enjoyed the fact that after some years of wearing the quite thick plate, I found I couldnt speak clearly without it and without lisping and tripping over my words.
I took it out one time, just before going into someones house to talk to them. It was so difficult that I had to make a quick excuse and go back out to the car and put it in again.

Did I explain that well? I am not sure.
Sorry if I have said it all before.
Anyone else think the same way?