Back in the 90s, when I watched Lois & Clark: The New Adventures of Superman (the Dean Cain series), there were a few different ways for Clark Kent to change clothes into Superman. One method involved him stepping into a room, closing the door, then immediately opening it again to come out in different clothes. That's basically what Brachetti does, as a "quick change artist", and it's very impressive: he'll be hidden from view for a very short space of time (sometimes less than a second), and change into a different outfit. There's a video on his website which shows a few examples.
In a way, this is similar to what Derren Brown does: he's addressing the audience directly (rather than us being invisible voyeurs to a story), and there's an element of illusion involved. However, he's not really a conjurer, and people aren't scratching their heads trying to figure out how he did it. The changes are hidden from view, but presumably the clothes are done up down the side with velcro straps (or a quieter equivalent) and he has assistants to yank things off.
It's a relatively short performance (1h 45m including interval), as compared to other plays I've seen, but I don't think it should have gone on for any longer; as impressive as it was, it's basically the same thing being repeated several times. Actually, I think my favourite part of the performance was when he did shadow puppets, which made a nice interlude to all the costume changes.
At the start of the performance, when they reminded everyone to turn off mobile phones, the announcer referred to Mr Brachetti's "advanced age". He came on wearing a grey wig, and then a lot of the show involved him switching back and forth between this character and his younger self. I assumed that this was misdirection, and that he's actually quite young while pretending to be old. However, he was apparently born in 1957! If I'm that athletic when I'm 50, I'll be happy.
I was sitting at the back of the first floor balcony. (Well, first floor relative to the stage, although this was slightly below ground level relative to the theatre lobby!) I had a good view, and I only had to duck my head once (to see something at the very top of the stage), so I assume that most seats in the the theatre would be equally good.
Here are a few other reviews: The Times, The Guardian, The Telegraph.