I don’t understand what it means to “weaken the argument.”

An argument will have premises -- facts and background information -- as well as a conclusion based on those premises. There are unstated assumptions that are necessary for the premises to logically lead to the conclusion. If these assumptions are true, then the conclusion is strong; if one or more of these assumptions turn out to not be true, then the conclusion is weakened.

In a "weakening" question, the correct answer choice will weaken the connection between the premises and the conclusion. Most likely, the correct answer will show how an assumption that we need for the conclusion to be strong is NOT true.

For more information, watch this lesson video on Weakening Arguments. Then, you can learn more about this in this GMAT resource, which in this case applies to GRE as well. 

I don’t understand how (B) weakens the argument.

Let's look again at the conclusion, first.

Until the drought ends, Milanda will continue to import more than it did before the drought started.

In short, our conclusion is that the increase in imports is happening because of the drought. If we get any information that tells us the increase in imports is due to something else, then we will weaken that argument.

(B) gives us that "something else." Milanda's economic wellbeing rests on imports from many different industries.

So it would be wrong to assume that the import level is directly related to the continuation of the drought. Say the Milandians during the drought want the recently released iPhone. Imports will increase, though these imports will have nothing to do with food. This reasoning works in the other direction as well: Maybe the number of imports of another non-food item will go down, and even though the drought continues, imports will have decreased.

I think that choice (D) weakens the argument. Why isn’t it correct?

The passage argues that because of a shortage of crops due to drought, Milanda will have an elevated number of imports until the end of the drought.

Choice D reads: "The amount of imported food that the average Milandan consumes has increased since the beginning of the drought."

This is essentially just a restatement of what the passage has already told us: because there is a food shortage, Milandans have to import more food. Therefore the average Milandan is consuming more imported food since the drought started.

It does not call into question the argument, which is that Milanda will import more things in general (not necessarily just food) because of the drought.