First, what you don't do is call the Emirates reservations number. They, in my experience, will send you on a wild goose chase where you make multiple international phone calls during Middle Eastern business hours into the phone system of Arabian Adventures, the Emirates-owned tour operator. Each agent that you're told to speak to will seem completely baffled by your request and unsure why you are asking them.
Instead, you go to this web page belonging to Kompas Express, the company Emirates has contracted to organize such packages for travelers in the US and Canada. I am aware that that URL looks like a scam, but it is linked to by this page on the official Emirates website intended for travel agents.
Go to the Kompas Express page, press "Go," and fill out the form to supply your booking details. An agent will then email you with some preliminary price quotes based on the options you selected. You can then write back and ask for prices for other hotels besides the ones they give you or to request other services. They'll put it all together in a quote and allow you to pay through a secure site. It is not possible to communicate with them in any way besides email.
You can, of course, book your own hotels separately, but then you won't get the transportation, 24-hour check-in/check-out, breakfast, or other parts of the package, so it's worth shopping around.
Also note that some itineraries qualify for Emirates Dubai Connect, where they will provide free accommodation and meals to passengers on qualifying fares and itineraries with long connections. This question is referring to voluntary long layovers for the purpose of seeing Dubai, not involuntary ones due to flight schedules.