Several reasons, in descending order of importance:
a) I don't have anywhere to store a bike in my apartment. My work has a bike room, but it's less convenient to use than the Citibike station right on the street. My roommate used to keep her bike locked up on the street at our apartment, and it got stolen only a few months after she'd gotten it.
b) Even if I owned my own bike, I might still have Citibike anyway, because there are trips that I make (usually at night) where I use Citibike to go meet up with people somewhere, and then we continue traveling using other modes of transit, such as cab or subway. Being able to leave a Citibike behind is really nice.
c) Maintenance. I don't have to worry about it at all beyond pressing a button when returning a Citibike that's misbehaving. There's a non-trivial amount of maintenance involved in keeping a commuter bike running (especially one that's ridden in rain and snow), and I just don't have to worry about it at all.
Most of the people I know who use CitiBike use it as one leg of their commute. One of my coworkers, for example, takes the PATH in from New Jersey then bikes to the office from there.
In these cases you could buy a bike and leave it in the city, but there's a very very serious risk of bike theft. It's easier to pay for the CitiBike membership and not have to worry about it.
I wish someone took economy of scale and produced a resilient bike tracker so cheap it could be embedded in the factory. Bike theft is the main reason I don't use my bike anymore (sic).
I take BABS (Bay Area Bike Share), the equivalent system in and around San Francisco, which is similar in nature to the Citi Bikes program. In fact they use the same bike model. I have 2 bikes of my own, but it's more convenient to hop on any random bike and drop it off without having to bring a lock. Sometimes I also want to go somewhere to meet people and then we walk or take the train/bus back. I live in SoMa, where most stations are, and I rarely walk anywhere these days since BABS is more convenient, and more fun. The only complaints are that all stations are around SoMa and the financial district (being addressed soon) and that the bikes are too heavy, about 40lbs which is a problem in a hilly city.
Discovering this was the first and last time I used the bikes in Montreal. It didn't help that the bike station was at the bottom of one of the steepest hills in downtown MTL. It also didn't help that two of the first bikes I selected were broken and had to be returned for another.
In New York, no one has space to keep a bike in their apartment. Okay, that's a bit of an exaggeration, but it is an issue. It's certainly an issue for many here in Boston, where Hubway bikes are popular. I ride my own bike to work, but I have a shed I can keep it in.
