I was confused like you too before I did it and couldn't understand how I would feed my son whilst he was still asleep...and also ensure that he did stay asleep! But it does work trust me.
If you are breastfeeding the best thing to do is firstly just hand express a little milk from the breast you want to start with onto the nipple. Then pick up your sleeping baby gently and bring his lips to the nipple and just gently touch his lips...he will automtaically smell and taste the milk and should start to suck as an automatic reflex. The only thing you may find is because he is still asleep or at least very sleepy it may be a slower suck than usual and so may take a while. I found a way around this and each night, after putting DS to bed at 7pm, I express at about 9.30-10pm each night. I then give him this milk in a bottle at 10.30pm. Again I just pick him up gently and sit him half propped on my lap and just gently place the bottle in his mouth...again he just instinctively starts to suck. If your baby is formula fed you can do this the same way.
I don't burp him at all. I just wait five minutes for the milk to be digested and then gently place him back into bed. He stays asleep pretty much 99% of the time. If he does awaken it is only temporarily and by the time he has finished the bottle he is asleep again.
It does seem strange and scary at first but you will find it easier each time you do it. I also wrap my son in a tight swaddle wrap each night, and feed him wrapped, and this helps keep him asleep also I think.
Even though during the day I always burp him, and often he will regurgitate his milk after each day feed, I have never had any problems with not burping him at the dreamfeed and he has never brought the milk up either at that feed. They say you don't need to burp them at the dreamfeed as they don't take in much air when they are sleepy.
Hope this helps you...good luck. Feel free to ask me anymore questions. It is worth persisting. Since doing this from 12 weeks onwards my son now sleeps from 7pm until 5.30-6am each night, with the dreamfeed at 10.30pm. He is only 17 weeks old so I am very happy with this result! Also if you express and give the milk by bottle sometimes hubby can do the feed so you can get more sleep....bliss!!