In the arrivals hall, before immigration, on the far right hand side, you will find the Dubai Connect desk (you cant miss it, its huge). They will issue you a voucher and give you details about how to get to the hotel etc.

As a US citizen you do not need a visa. Your passport will simply be stamped and you will be waived through as a tourist.

When you check in to the hotel, youre free to do whatever you want for the duration of your stay, you're effectively treated as a normal tourist/guest as in any other city.