Jack O’Connell, the popular yesteryear actor known for his comic role

An inference is a conclusion that we draw based on what we read in the passage.
The conclusion we draw from the facts must be true.
The conclusion must be something that undeniably and naturally results from the facts.

This question tests whether you know what an inference is (and, I suspect, whether you are not rattled by the question).

This answer is much too strong, a characteristic that is typical of wrong answers on inference questions.
We have no idea whether option A is true, let alone whether it must be true.
One person's concern about his particular history cannot be generalized into this kind of sweeping statement.
Eliminate A
We have no idea whether this statement is true.
Irrelevant. The prompt never mentions anything about winning candidates or successful mayors.
Eliminate B
Much too strong. Resembles (A), except (C) is worse: it extends to politics generally. Eliminate C.
Careful. This answer is a trap.

EDIT: It's a trap in one of two ways:
(1) it changes "only if" to "if," a logical move that does not hold
or
(2) it restates a premise.

I think the author could have intended to create either issue. (And A, B, and C are so horrible that I'm not sure about the level of sophistication that this question displays.)

(1) Only if and if

Valid: He will run only if he is confident.
Valid: IF he runs, then he is confident.
Also valid (contraposition): If he is NOT confident, then he will NOT run.

In logic, we would write
If R, then C.
If not C, then not R.

But what if he does not run? Is he not confident? We have no idea.
We cannot conclude anything. Confidence is a necessary but not sufficient condition.
He could be confident and still decide not to run. Maybe he goes back to his TV show.
Valid: If R, then C
NOT valid: If not R, then not C

What is he IS confident? Will he run for sure?
We have no idea.
He must be confident in order to run, but the fact that he is confident does not guarantee that he will run.
Confidence is a necessary condition, not one sufficient to guarantee that he runs.
Valid: If R, then C
NOT valid: If C, then R

Option (D) looks likes this fallacy immediately above: IF C, then R.
Wrong: If he is confident, then he will run.
We don't know for sure what will happen IF he is confident.
He might run. But he might not run.

The necessary condition created by only if means that IF he is confident, he might contest the election (run for office).
He could run. But he might not do so.

Valid: If R, then C.
Valid: If not C, then not R.

Incorrect: If not R, then not C.
Incorrect: If C, then R. (Option D looks like this invalid statement.)

(2) Alternatively, Option D does not show the difference between IF and ONLY IF, but rather, restates these words:
he would contest only if he feels confident that his impression as a comedian will not adversely affect the public’s confidence in his abilities as a Mayor.

Restatement of a premise is not an inference.
We must reason to a conclusion based on the information in the passage.

Restatement does not require reasoning.
Inference requires reasoning.
D either wrongly changes ONLY IF to IF, or incorrectly restates a premise.
Eliminate D
This is the answer. Reason backwards.

"If O'Connell becomes the new mayor" = O"Connell won the election for mayor.
He got elected.
Obviously, he ran for office.
And in order to run for office, according to his own standards, he believed that his comedic history had not hurt the public's confidence in his mayoral abilities.

IF he ran, he was confident.
He did run. He won. He was confident.

We have to stretch a little to reach the inference in (E).
We are not sure whether the public actually had confidence in his abilities.
But we are sure that he ran because he believed that his TV role had not hurt the public's confidence in him.
Close enough. And a lot closer than any other answer.

The answer is E.

I hope that helps.

The extremely talented JonShukhrat alerted me about (D) after I posted.
He believes that there is only one problem with (D), namely, that it changes ONLY IF (necessary condition) to IF (sufficient condition).
He does not believe that (D) repeats a premise.
I am not sure. (I am not sure which of the two mistakes the author intended. If this question were official, I would agree with JonShukhrat that the error derives from only if and if.)
I think that JonShukhrat should explain how he sees (D).
What say, Jon?