Favoritethings gives a good explanation. I would just like to bring to light a couple of ambiguities. The gas attack in this poem shows characteristics of chlorine gas, alongside mustard gas. Mustard gas tends to be slower acting and produces blisters. Chlorine gas produces froth in the lungs and asphyxiation and watering, irritated eyes, as described, and the "thick green light" in the poem is consistent with chlorine's color. Chlorine gas was used first, then phosgene gas, then mustard gas. Mustard gas was first introduced in September 1917 against the Russians. Owen's poem was penned as early as October 1917, so the timing of the writing is too close to be able to definitely say whether it is chlorine or mustard gas that is used in the poem.

The speaker says that in an "ecstasy of fumbling," the soldiers race to put on their gas masks (9-10).  World War I was the first war in which mustard gas was used as a weapon.  If a man inhaled this gas, his lungs would fill with blood and he would die, essentially by drowning in this fluid. 

The man who didn't get his "clumsy helmet" on in time inhaled mustard gas, and the speaker (with his gas mask on) watches him flail and contort as though he were being burned "like a man in fire" (12).  The helpless narrator sees this doomed man through the mist of the gas, and describes him as "drowning," because this is basically the kind of death the gas causes (14). 

The other soldiers "flung" his body into a wagon, and he must have remained alive, suffering, for a while because his eyes were "writhing in his face" and bubbles ran out of his mouth as a result of his "froth-corrupted lungs" (19, 22). It took some time for him to die, and it would have been an extremely painful, awful death.  Now, the speaker says, he sees this man in his dreams, choking and dying all over again. He says that if we, too, could see this man who died after inhaling mustard gas, we would never again say that it is sweet and fitting to die for one's country.