Can someone who does attachment parenting explain what it involves, why they do it and how it has benefitted them and their children?
I'll try...
what it involves
I think the best description I can do is the overview here from Attachment Parenting International, which outlines the general philosophy. In very, very brief, it's an approach to parenting that focuses on responsiveness to a child's needs as the basis of promoting secure attachment.
A lot of the practices associated with AP come out of Bowlby's work on attachment in the 1950s and on, which say (again, in very very brief and I'm generalising here) that what helps attachment in young children is what they're evolutionarily designed to expect: a lot of close physical contact, quick responsiveness to crying, etc. So people who do AP with babies will be a lot more likely to carry them in slings a lot, feed them on demand, and not leave them to cry.
Many people will tell you that attachment parenting says you have to breastfeed, have to cosleep, have to give up work and be a SAHP. But really, it's not a list of rules, it's a general approach with a bunch of tools people pick and choose from to support that approach. I do some things people associate with typical AP (breastfeeding up to toddlerhood, co-slept) and don't do other things (I work full-time, I have no plans to homeschool).
why they do it
Because I agree with the general philosophy (responsiveness is important for attachment, modern Western parenting practices often don't suit the evolutionary needs of children). Because it suits me and my family.
and how it has benefitted them
As above, it suits my general approach to parenting, and it sits a lot better with the way I intuitively like to parent than, say, Gina Ford does. So that's a benefit.
Another benefit is that, since my child was a very tiny baby, I found attachment parenting approaches a lot less guilt-inducing than other approaches. My baby wanted to be held all the time, hated prams, loved slings, wanted to feed constantly. I had a lot of people (friends, family, books, HV/midwives) saying that I needed to train my baby not to do this - that my baby was snacking, getting into bad habits, I was creating a rod for my own back, she'd be horribly clingy, etc. The more AP people/books said that responding to my baby the way I and she preferred was fine, and she'd grow out of it in time.
AP approaches have helped me understand my child's perspective on things, and have helped me take a more thoughtful, considered approach to parenting than I would have had if I'd just reflexively gone with the way I was parented. (Which wasn't awful by any means but wasn't ideal in every way, either.)
and their children?
Well, my only child's still just a toddler, and besides I can't really say "she's like this because I parented this way and she'd be a serial killer if I hadn't!". But, she's happy, she's thriving, she seems pretty secure and independent, and she doesn't need to be carried round all the sodding time any more, glory hallelujah, so something's worked. And she isn't a serial killer so far 
And can I ask if the other people you raise your child with also use attachment parenting - parents, siblings, friends, partners, professional child care providers?
Partner - yes, we value being on the same page when it comes to parenting.
Parents - I don't think they would call themselves "attachment parents" (and in fact thought a lot of the things we did when our DC was little were barking mad, although they've generally come round on that!) but they take the same kind of general approach we do when caring for DC - how to approach tantrums, and so on. I don't think it's essential for everyone who cares for a child to do things the same way, but I do think it's important that they're on the same page in general.
Professional childcare providers - same kind of thing. We didn't go out of our way to find a childcare provider who identified themselves as an AP follower, but we did turn down the nursery who dealt with tantrums by using time-outs from the age of 12 months, because that wasn't the kind of approach I wanted my childcare provider to follow.