To choke or cough as a result of something irritating or blocking the throat/airway; to be stifled by (smoke, food, tears, etc.).
魚を丸ごと飲み込んで喉に詰まり、むせてしまった。
He swallowed a whole fish, it got stuck in his throat, and he started to choke.
たばこの煙でむせることがあるので、換気をした方がいい。
Cigarette smoke can make you cough, so it's better to ventilate.
唐辛子を食べてむせそうになった。
I almost choked after eating chili.
涙でむせて声が出なかった。
I was choked up with tears and couldn't speak.