That's how I used to think, too - I really hate the idea of having a monthly payment for something I don't own in the end. Two things led me to consider it: One is a coworker of mine who is quite savvy financially who is now leasing his vehicles (I guess that's what got me to consider it at all)...
The other is that I'm in a situation where I need a new (to me at least) car. In the course of researching various vehicles, it seems like every single one I've looked at had a model refresh in 2013 or 2014 - Probably a side effect of the Great Recession and lowered R&D budgets at the manufacturers. But, I don't want to get a 3-year-old car that already looks like it's 10 years old. In addition, being a fan of efficiency, I'm looking for modern powertrain options that have only come into existence in the last 2-3 years and are very difficult to find on the used market.
Specifically, I like the Ford Fusion Energi - It's a plug-in hybrid with the range to get me to work and back with little or no gas burned. I took a test drive the other day that was about the length of my round-trip commute and burned 0.14 gallons of gas for an average of 158 mpg! (Whether the cost differential between the gas I would have used and the electricity to charge it is worth the higher initial cost remains to be calculated.)
I'm of the "I drive old cars so I can fly old airplanes" camp, but I just don't feel like that's gonna work this time, and money is so cheap right now I'm looking at all my options. I wish I could find a used Energi but they're in high enough demand (and of course low supply) that they're going for almost what a new one would cost anyway.
The monthly payment on a regular loan is a bit stiff for my tastes right now, especially since our other car is getting long in the tooth as well and we're likely to end up with two car payments soon. But, the numbers on the leases look favorable, so my likely strategy would be to buy the car at the end of the lease. That would allow me to stretch the payments across a 3-year lease plus maybe a 5-year loan at a reasonable rate, whereas the rates on a new-car loan greater than 6 years start to become less reasonable.
Now, tell me why that's a dumb plan - I certainly wouldn't go in expecting to just turn the car in and get another one in three years.