In 1578 Moroccan troops defeated a Portuguese army on Moroccan soil.

What the argument says:
1. Moroccans defeated the Portuguese and killed the king, but there is no mention of the killing of the king in history
.....a lot of factual information/story.....
2. The omission of the killing of the king has a psychological cause- the Portuguese found Sebastian's death so humiliating that they could not write about it

What the question asks:
We need to identify the significance of a part of the argument.

Strategy:
For this type of question, the best approach is to identify the significance of each sentence from the start. Let's proceed this way-
In 1578 Moroccan troops defeated a Portuguese army on Moroccan soil. - This sentence provides the base of the rest of the argument. It provides information about an event, which is possibly later discussed in the argument
Accounts written by Portuguese contemporaries report the defeat but omit mention of the fact that King Sebastian of Portugal was killed in the battle. - This sentence signifies an 'outlier behaviour', i.e. something which was expected to happen, didn't happen
Such omissions cannot simply be the result of ignorance of Sebastian's death. - Suggests a possible direction the rest of the argument would go towards - identifying why there are omissions of the king's death
Sebastian's death is not even mentioned in the battle accounts written by two captured Portuguese officers while waiting to be ransomed from a Moroccan prison. - This provides an additional surprising story to wonder about the omission of the king's death from history
These two officers shared their cells with the Portuguese soldiers who identified the king's body to Moroccan officials. - Interesting! The argument now tells about the two soldiers. I wonder why!?
[b]The omissions, therefore, had to have had a psychological cause: the Portuguese evidently found Sebastian's death so humiliating that they could not bring themselves to write about it.[/b] - Nice! So, finally, we are where we wanted to be - identifying the reason for the omission of the king's death from history. The author says that the omissions HAD TO HAVE HAD a psychological cause. That's quite a strong stand!

Now let's move on to the answer choices-
(A) supplying indirect evidence that a disputed death actually occurred - An "evidence" means a fact or a sign of the truth that makes us believe something. This answer choice indicates that the story about the two prisoners is presented because the author wanted to present us with some sort of evidence. However, there is no evidence, just a premise (the two prisoners were in the same cell) and a conclusion (the omission has had a psychological cause). Hence, this answer choice could be ruled out
(B) resolving an apparent contradiction between two of the argument's main premises - Identifying the premises is the key here. The argument's two main premises (1. The Portuguese lost the battle, and 2. Omission of the king's death in history) together offer a contradiction. However, this answer choice indicates that the two premises contradict each other.
(C) giving evidence supporting a general psychological principle on which the main conclusion is explicitly based - This answer choice suggests that the story about the two prisoners gives evidence that supports a general psychological principle. Is it so, though? We are not sure. Maybe the argument is made by a history enthusiast/expert or a psychology enthusiast/expert or none of the two. If it is made by a psychology expert, one could suspect that the conclusion is based on some background theory and experiments that are not mentioned in the argument. However, we are not told that the argument is made by a psychological expert. Hence, this answer choice falls short
(D) providing groundwork for eliminating an alternative explanation - This sounds in line with what we discussed in our strategy. The author feels so strongly about the cause that he has mentioned in the argument that he considers it to be the only possible explanation (The omissions, therefore, had to have had and evidently). Let's see what the other option has to offer. Unless it has any better suiting answer choice, we can be sure to select this
(E) offering grounds for doubting the reliability of historical reports that conflict with the argument's main conclusion - This is an easy no-go. If anything, the argument supports the reliability of the historical reports

Hence, the correct answer choice is D