The main reason for strictly using "большой опыт" in this case is that there are no further qualifiers as to the nature of the author's experience; and using "много опыта" without indicating what exactly the subject of the sentence is experienced in sounds somewhat incomplete.

Basically, this is just one of the many common ways of saying that the author in question is an experienced author, full stop.

If we were to change the sentence to something like 'This author has a lot of experience in writing poems', then you can use either:
У этого автора большой опыт в написании стихотворений.
У этого автора много опыта в написании стихотворений.

Also works if you're referring to something mentioned earlier, like if someone asked you if the same author could write a poem for their daughter.
Да, у него в этом большой опыт.
Да, у него в этом много опыта.

The two are generally understood to mean the same thing, but "большой опыт" does indeed connote some kind of a holistic skillset developed over a long period of time by means of rigid practice, whilst "много опыта" connotes experience gained by having done the thing in question many times in the past.

I second that, as it sounds like nonsense to me which was just literally translated. My first assumption was that this is simply a translation error. Despite the difference in meaning in Russian, большой опыт and много опыта should probably both be translated as "a lot of experience", "much experience", or something along those lines.

"Broad experience" is used (at least in Australia) to mean something like the holistic skill set described in a comment below - perhaps this would be a better translation. I agree that "large experience" sounds very clumsy.