With an extended warranty, that provides the proper coverage, you are basically insured against all sorts of nasties that could wrack up thousands of dollars in repairs. In the long term, this can save you more money than if you pay out of pocket for those repairs.
The peace of mind you get from knowing that you are guarded against that rainy day should not be underestimated. It is human nature to believe that the long shot bad luck will never happen, only it is almost inevitable that it eventually does happen down the road. An extended warranty covers that long shot for an additional period of time, hopefully until you are done with the car. No one likes nasty surprises.
Extended warranties make very decent bargaining chips at the table if you decide to sell the car. Of course, the warranty has to be transferable; it is something you should take into consideration when you purchase the extended warranty in the first place.
The perks that come along with some warranties are icing on the cake: covered rental, towing and possibly discounted maintenance.
You car does not break down during the period or duration of the extended warranty. Money wasted.
Warranties are a lucrative business for any company selling them. This suggests that ultimately they pay out a lot less in repairs than they take in. Extended warranties cover additional time/miles where breakdowns are more likely to occur, and are proportionally more expensive because of it.
Nevertheless, companies still feel extended warranties are a profitable business or they would not keep offering them. This tells you something about the odds.
More hassle at the contract table trying to figure out what every last word in the contract means and just how deep your coverage will be and how much you will have to pay if you sign the dotted line. No doubt, there are plenty of warranty companies who deliberately word contracts to leave loopholes the size of barnyards for them to wriggle through. Be sure to double check a company's reputation at the Better Business Bureau or Consumer Reports for comments and ratings from other consumers.