Dear jlgdr, kinghyts, and shagalo,
They mean opposition to the first "section in bold face".
The political analyst suggests that America needs a third party. This is the main point of the analyst's argument.
The analyst first points out an "opposing claim" made by some people. This is the first bold face section. The analyst then points out an additional claim made by these people.
The analyst responds to these claims by providing "facts" (evidence) that counter the first claim (only the governance one, not the additional claim of the sluggish pace of government and the burden of a third party).
Put all these elements together, and you get D. D clearly explains the functions of the two bold face portions.
What about B and E?
B says that the first opposes the premises of the argument. What are the premises? Multi-party systems seem to be doing rather well -- a fact. The first bold face doesn't say that multi-part systems are NOT doing well; it merely upholds a different claim about the two-party system. It is perfectly acceptable for both two-party and multi-party systems to do well; one doesn't exclude or preclude the other. So the first bold face does not oppose the premises of the argument. B also says that the second is a claim that the argument supports. One can verify whether most countries have multi-party systems and whether few/many people complain about the governance. So the second bold face contains two verifiable statements, i.e. fact/evidence, not a claim.
The first part of E is correct; the first bold face portion is indeed a claim made by opponents of the main point of the argument. But the second is not a claim, as discussed above. Also, the first does not oppose the fact/evidence in the second part, as discussed above. (Another subtle point is that, if the first opposed the second, it should have come before, not after, the second, in the sequence of events.)
Evidence can effectively oppose a claim; it's absurd to imply that a claim can oppose fact.
--Prasad