When you hit the check-in counter at your airport of departure, make sure you ask for your voucher for Dubai Connect. Then, upon arrival in Dubai, Emirates airline personnel will guide you to your ground transfers, and away you go. Once at the hotel, present your voucher for your free night. When booking your stay, double-check that a meal at the hotel is also included.
You can book the Dubai Connect option up to 24 hours in advance of your trip. But it is, of course, subject to availability.
Now, about that hotel...
Don't get too carried away there, buddy -- you're not getting a free night at the Burj al Arab. The hotels are along the lines of ones US airlines use to put passengers up when flights are cancelled. That is, reasonable, comfortable hotels within a short drive from the airport.