Why do I start to cough when I eat or drink cold things?

Cold air. Breathing air past cold items can cool the air. Cold air can make many people cough. Sometimes people cough because they swallow improperly, but that has nothing to do with temperature. If you have no other symptoms, don't worry about it.
Could be asthma. There are many things that than cause after eating. One could be aspiration- accidentally swallowing liquids down the wrong pipe. If this only happens with cold, there is a phenomenon called bronchospasm, a component of asthma where the tubes in the lungs tighten up in response to cold which can cause a need to cough. This can be tested for and treated by a physician.