In the early 1920s, a time when new technologies were revolutionizing

OFFICIAL EXPLANATION

This question is a much easier variation on a question that I posted not long ago in Butler. (The same author wrote both questions.)
Strip the sentence and condense as much as possible to make POE easier:

In the early 1920s, a time when new technologies were revolutionizing leisure activities, the music business became a major sector of the entertainment industry by producing XYZ millions of dollars worth of goods and it began to exert a strong influence on popular culture.

The sentence, stripped:
In the early 1920s, the music business became a major sector of the entertainment industry by producing XYZ and it [?] began to exert a strong influence on popular culture.

POE

• Split #1: IDIOM - begin, began, beginning

-- When begin is in verb form (not present participle), and the verb tense is not progressive (continuous), the verb can take either an infinitive or a gerund.

Correct: She began to speak.
Correct: She began speaking.
Wrong: She was beginning speaking.
Correct: She was beginning to speak.

-- The last two examples above fit with this next idiom:
if begin is in a continuous (progressive) tense, then begin can take only the infinitive.

-- beginning is a participle (verbING).
Whether beginning is used with a helping verb (e.g. was beginning) or as a participial modifier (a verbING), beginning can take only the infinitive.
Correct: The plant, finally beginning to grow, needed more water than I had been giving it.
Wrong: The plant, finally beginning growing, needed more water than I had been giving it.
Wrong: beginning that

Options C and E incorrectly use beginning that.

Option B incorrectly adds the preposition IN between began and exerting.
-- There is no such thing as began in exerting
-- The correct construction is either began to exert or began exerting.
-- The verb phrase began in exerting is both unidiomatic and nonsensical.
-- Began BY exerting might be okay, but not in this context. Wrong meaning.

Eliminate B, C, and E

• Split #2: Compound predicate guidelines

Option A needlessly inserts the pronoun IT before the second verb of the single subject "the music industry."
Option D does not include IT and thus is better than Option A.

When two verbs share one subject, we have a compound predicate.
Stated differently, when a single subject is followed by two verbs, we have a compound predicate.

A predicate is just the verb part of the sentence (everything that is not the subject, including the subject's modifiers).

In remaining options A and D, we have
One subject: the music business
Two verbs: became [a major sector] and began [to influence popular culture]

When one subject shares two verbs, almost always, do not repeat the subject before the second verb
-- We want to keep subjects and their actions (their verbs) as connected as possible.
For that reason, typically we do not put a comma between a subject and its second verb.
Also for that reason, typically we do not "repeat the [single] subject" by inserting a pronoun before the second verb.

-- Option A inserts the word IT.
Now the reader has to stop. Is this IT different from the "music industry"? The reader checks.
No. It and the music industry are the same thing.

Option D does not interrupt:
[T]he music business became a major sector of the entertainment industry . . . and began to exert a strong influence on popular culture.

The form in option D is preferred because that form is usually clearer and more forceful.
-- If repeating the subject makes the sentence much clearer, repetition is okay.
That situation nearly always happens when the first verb phrase contains a lot of commas between descriptors or a lot of commas because it contains a list.

Some people describe option A as a run on.
You can call the structure in (A) whatever you'd like.
Just understand that a subject that does two things should stay as connected as possible to those actions.
A comma or a repeated subject or both before the second verb generally create disconnect between the subject and its verbs.

The pronoun IT is unnecessary.

For a thorough treatment of compound predicates and what to do with commas or repeated subjects or both,
see my post about an official question, HERE.
Spoiler alert - the link takes you to a post that explains some official answer choices.
That official question is about Joan of Arc.

PRONOUN ambiguity? Interesting.
I think that the logical antecedent of IT is "music industry."
At the same time, I cannot make an ironclad case against "the entertainment industry" as an antecedent.
My position: I would not eliminate (A) on the basis of pronoun ambiguity alone.
We can say, though, that (D) avoids both subject repetition and any doubt about what is doing the second action.

Eliminate A.

The answer is D

• OTHER ISSUES in options C and E

This sentence is incomprehensible.
I cannot rewrite it.
This effort is terribly flawed: "The music business became a major sector . . . by producing millions of XYZS and [by] beginning a strong influence on popular culture [that] was exerted."

Even if you do not know that beginning that is unidiomatic, the structure of the sentence alone should send you running away. Nothing is clear.
Alternatively, compare to (D). Cripes, compare to (A). Both are better than C.

This option also creates an incomprehensible sentence.
I have no idea what the word "should" is doing with the exception of aggravating me.
"Should have" is not warranted.
-- We can use should have in some predictions
Correct: The medicine should have a palliative effect within hours.
-- We can use should have in sentences that express regret, disapproval, or "advice."
Correct: The artist should have ignored her critics. (Critics are just people with opinions. )

Options C and E are awful.
If you get stuck, give yourself 10 seconds to make a convincing argument that C and E are better than D.
(You could even compare these two disasters to option A—unless you are really keen on pronoun ambiguity, a keenness you should probably soften considerably.)

The answer is D

COMMENTS

I like this question because the idiomatic structure used for beginning/begin is tested occasionally. The idiom is not well-known.
Because the idiomatic structure is not well-known, we might be forced into a tough spot.
-- Are options C and E somehow parallel to "by producing"?
-- Parallelism can be daunting. The urge to understand the matter perfectly is understandable.

But that urge often is counterproductive.

Many of you correctly noted that options C and E simply do not make sense.
-- If you cannot "rethink" a sentence's meaning within 15 seconds, eliminate it. (I would not give an option that long. I'm accounting for inexperience and nerves.)

If I have another option that is clear and contains no errors, I discard the offending options after about 3 seconds.
Comparison is part of POE.
After a certain amount of practice (and reading in English!), trust your instincts when you compare.
Which is best: C, D, or E?

That answer is not a hard call if you trust that part of POE is comparing bad answers (worse) to good ones (better).
Choose "better." Not perfect. "Better."
Psychologically, though, making that call CAN be hard. That's okay. You can fix the hesitancy. Take a breath and practice comparing options in questions in this setting.

A few errors exist here and there in the explanations. That said, those who actually explained did a nice job.