How can exercise trigger asthma?

Cold air. Exercise leads to mouth breathing because when you exercise you can't get enough air through your nose. The nose warms and moistens air; the mouth not so much. Cold, dry air irritates the cells that line the breathing tubes that in turn release biologic mediators that cause airway muscle cells to contract narrowing the breathing tubes causing coughing, wheezing & chest pain & tightness.
More demand. The simplest way to put it is that there is more demand on the lungs to exchange oxygen and carbon dioxide during exercise. That requires faster and deeper respirations and may strain all parts of the lungs.