What exactly happens during an asthma attack?

Attack... During an asthma exacerbation, the inflammatory cascade in the airway is triggered leading to bronchospasm and airway inflammation. The airways become swollen and fill with inflammatory by-products like white blood cells, mucous, etc. Treatment is aimed at relieving the bronchospasm and inflammation.
Airway inflammation. The smaller air carrying tubes in the lung become inflamed and the muscles around them spasm. This leads to air getting trapped out in the lung, so it becomes hyperinlated. As a result, the sufferer can not get air out. .. Or in. If severe enough and untreated this can lead to death.
Constriction & mucus. An asthma patient has episodes of bronchospasm (airways tighten up) plus swelling of the airways & increased mucus, causing difficulty breathing and creating wheezing noises. Inhalers like albuterol relax & open up airways; inhalers like Flovent (steroid) prevent or decrease swelling & mucus. Many children outgrow their asthma, so are almost "cured" but may get an episode years later.