Just as "only if" indicates a necessary condition, even though it uses the word "if", which normally indicates a sufficient condition, "only someone who" indicates a necessary condition. The "only" makes the whole phrase necessary.
Therefore, the conditional in the conclusion is:
someone has a chance of being elected
that person truly understands economics
It looks like you were on the right track in seeing that the conditionals in the premises create some double-not arrows:
supports the new tax plan
has a chance of being elected
supports the new tax plan
Since these are double-not arrows, the only thing excluded is that the things on each side are both true. There's nothing that says they can't both be false. So, for instance, someone could fail to understand economics AND fail to support the new tax plan. The conclusion says that anyone who has a chance of being elected must truly understand economics. By the double-not arrows, we can see that someone who has a chance of being elected cannot also support the new tax plan. But if they don't support the new tax plan, do they need to understand economics? No; in the second double-not situation there, someone can fail to understand economics and fail to support the new tax plan. And that is exactly what answer choice (D) says.
Diagramming helps because, once you've made explicit what things can't go together, and why, you can easily check the possibilities that are still open, and see why the conclusion doesn't have to be true. Now, it's hard to do this if you're not sure what's the sufficient and what's the necessary condition, and this was troublesome for you in this question. That's why it's vital to practice identifying the sufficient and necessary conditions whenever there's a conditional in the stimulus. If the phrasing throws you a curve, like it did here (where it mixed keywords so you weren't sure which was sufficient and which was necessary), try to rephrase what's being said. Here, the conclusion (which was flawed!) claimed that a person who has any chance of being elected MUST truly understand economics. If you're not sure what's sufficient and what's necessary, ask yourself: if I have A, do I have to have B? If so, then B is a necessary condition and A is a sufficient condition. If not, then ask: if I have B, do I have to have A? Then A is a necessary condition and B is a sufficient condition. If neither is necessary or sufficient for the other, then they aren't directly connected. Look for chains of conditionals that might connect them.