I just moved to a new apartment myself, and these folks are exactly right. Definitely be sure to choose a small, quiet room that the cats will stay in for the first few days (or possibly several days) that they are in your new home. Make sure to move all the furniture ect. in first, then the cats.
Try to put as many familiar things in the room as possible. We kept our cats in the bedroom when we moved in last week. We set up the furniture similar to the way it was in the old apartment and used the same comforter on our bed as we did in the old place so it had our scent and our cats' scent on it already. Make sure you have some familiar toys out for them too. When you let them out in their new room, be sure to show them where the litter box and food/water is first thing, then let them explore on their own. They will probably choose to hide first, then come out and explore as they feel more comfortable.
It's a good idea to include a little bit of used litter in the in the litter box, because the smell of used litter will encourage them to use the box. It also helps to have a radio or tv on softly. The sounds make the cats feel more comfortable. Spend some time with them in the room, talking softly to them. Let them know you are there, but don't force them to come out if they hide.
It make take a day or up to several days for them to adjust. Once they are comfortable in the room you choose for them, you cat let them explore more of your apartment. My cats adjusted quickly. I was surprised that they didn't even hide after I showed them their box and food. After only two days they were reaching under the door with their paws, so I felt they were ready to explore more of the apartment. I closed the door to the other bedroom and bathroom, and allowed them to come down the stairs to check out the livingroom and kitchen. After another couple of days they were comfortable with these new rooms and I let them into the rest of the apartment.
Remember to put things in the room that smell like you and the cats, put familiar items in the room, organize the furniture in a similar manner to they way it was in your last place, and show them the litter box and food right away. Then let them do their own thing. You will know when they're ready to explore the rest of your apartment. Also, don't worry. I had some major anxiety about moving my cats, and everything turned out fine. Good luck and keep us posted!