Understanding the argument -
For most people, the left half of the brain controls linguistic capabilities, but some people have their language centers in the right half. - Fact.
When a language center of the brain is damaged, for example by a stroke, linguistic capabilities are impaired in some way. Fact
Therefore, people who have suffered a serious stroke on the left side of the brain without suffering any such impairment must have their language centers in the right half. - Conclusion
What if one suffers a stroke and it doesn't impact linguistic capabilities? Then, can we say that "they must have their language centers in the right half"? No, we can't. That is what option D highlights.
Option Elimination - We need to find a missing premise or a minimum condition that can make the conclusion work: "people (who have suffered a serious stroke on the left side of the brain without suffering any such impairment) must have their language centers in the right half.
(A) No part of a person's brain that is damaged by a stroke ever recovers. - "Recovery" is in no way linked to the scope at hand, which is finding a missing premise to make the conclusion work, which is "people (who blah blah..) have their language centers in the right half. Out of scope.
(B) Impairment of linguistic capabilities does not occur in people who have not suffered any damage to any language center of the brain. - People "who have not suffered any damage to any language center of the brain" are not in scope. We are talking here about people "who have suffered a serious stroke on the left side of the brain without suffering any such impairment." Distortion.
(C) Strokes tend to impair linguistic capabilities more severely than does any other cause of damage to language centers in the brain. "other damages" is out of scope.
(D) If there are language centers on the left side of the brain, any serious stroke affecting that side of the brain damages at least one of them. - Yes. Negate the main clause "If there are language centers on the left side of the brain, any serious stroke affecting that side of the brain damages "none" of them." or "If there are language centers on the left side of the brain, there could be a serious stroke affecting that side of the brain without damaging any of them." Then the conclusion "people (who blah blah..) have their language centers in the right half" shatters.
(E) It is impossible to determine which side of the brain contains a person's language centers if the person has not suffered damage to either side of the brain. - At best, it can be a weakener. Opposite of what we need.