I happen to have exercise induced Asthma, so I have asked a few of these questions. Here are some of the reasons that cold air causes you to feel like you have less usable oxygen.

The bronchial tubes in your lungs naturally constrict when their temperature drops. So even if you are not asthmatic you will still generally be taking in less air per breath than you would with normal air. A lot of body heat is lost during the breathing process, the cold air in your lungs has to be offset by warming the body so you are dumping extra energy into producing that heat. Trying to keep yourself warm also plays into more energy used since the outside of you body is attempting to stay warm as well.

So the biggest factor is not getting as much air into your lungs, and the a small factor is the oxygen by used to heat the body.