A derivative is a contract that is based on an underlying assets the value of which will fluctuate in response to the value of the asset. Derivatives are often used as hedges against risk in investments. In this scenario, the auto lease is a derivative contract against the underlying asset of the car. The derivative contract acts as a hedge (risk protection) against the depreciating value of the car: the lease/derivative limits the downward fluctuation in value thus insuring a reasonable sale price as a previously driven automobile.
This is a massive issue with the purchase of new cars, especially in the UK where from the moment you buy a new car at its market value you are thought to have knocked a significant amount off of the car's value. This is what the derivative value of the car concerns. It is therefore much wiser financially to buy a second hand car and therefore to make less of a loss by the depreciation.
Part of how a leasing company makes money is the resale value of its fleet of "slightly" used cars that come in at the end of the lease. The derivative is that the car will be perfectly maintained and paid for to more than cover its initial depreciation, and that they can then sell the used car for a profit in the used car market.
When you lease a car, you are agreeing to keep the car for a set amount of time, drive it only a little, and keep it in good condition. You do not actually own the car. You get the benefits of having a new car every few years, because you give it back, but you don't really own the car.
This is precisely why most lease agreements have mileage limits and penalties, to protect (hedge, in the context of this question) against excessive depreciation from the standpoint of the dealer.
I suppose that this agreement can be thought of as a derivative because it is based on the underlying value of the car. The person who enters into the lease believes that the car's value will depreciate by more than the amount he/she is paying for the lease.