You do not have to buy tickets for the length of stay to qualify to purchase the dining plan. You can if you want to.
Free dining is a promotion to encourage attendance and onsite stays at traditional low attendance times, specifically targeted at the height of hurricane season (mid-August through September), which scared off a lot of potential travelers after the seasons of 2004 and 2005. If you called Disney and purchased a package including an onsite room at regular rack rate, plus a minimum of a one-day base ticket, they would throw in the standard
Disney dining plan, originally priced at $39.99 per night of your stay for anyone over 10 years old and $10.99 for children under 10, at no extra cost. Everyone staying in the room had to have the same package and buy the same type of ticket.
I am not going to go into the particulars of the dining plan here, there are plenty of posts on the dining plan board that will help explain.
For 2007 it has currently been offered only to visitors from overseas and anyone who was able to book a trip while actually staying at a WDW resort in November and December.