Why is the sentence "She sighed, and began whispering again" grammatically incorrect?

I also do not see anything wrong with it but yes, there is no reason to put comma there. I think the comma is put by the author to denote a pause, but it is not grammatically correct according to the rules for comma usage.

You might want to see this: http://grammar.ccc.commnet.edu/grammar/commas.htm

I also think that sentence is perfectly okay.

It's also ( I know I have repeated 'also') an interesting construct because you do not grammatically require a comma, if the subject is the same but not stated, whereas 'She sighed, and she began whispering again, requires the comma because the subject 'she' is repeated in the subordinate clause.

However, I think this is a very good example of how English lets you play around with it, too.

Or if you like, the three sentences:

'She sighed, and began whispering again.'

'She sighed, and she began whispering again.', and

'She sighed and began whispering again.'

are all inherently different in their content and what their respective authors are communicating, and it is that that must be, in the final analysis, at least for me, the crucial deciding factor.

The comma is the putative mistake. Typically, students are instructed not to put a comma between two coordinated predicates:

She_subject sighed_{predicate A} [ and began whispering again ]_{predicate B}

For many people, "grammar" is a collection of every language-related prohibition they've ever learned (or think they've learned). Presumably they've internalized the rule above, and are excited to catch someone breaking it. That is, after all, how you play the game of Grammar Gotcha.

“When I saw him turning, I set my face towards home, and made the best use of my legs.”

-- Dickens, Great Expectations

“My proposal was to build a wharf there fit for us to stand upon, and I showed my comrades a large heap of stones, which were intended for a new house near the marsh, and which would very well suit our purpose.”

-- Franklin, Autobiography

It’s not difficult to find many, many more examples of this sort. If the OP is ungrammatical, it’s in good company!

(In other words, to be completely clear about it: it’s not.)

The comma in this sentence is perfectly fine.

If you were to take it out, you'd subtly change the meaning of the sentence:

She sighed and began whispering again

might imply that she sighed again and began whispering again. By inserting the comma, the writer makes it clear that only the whispering is the repeated act, that she hadn't necessarily sighed before.

It's a very minor distinction (and others may disagree). Comma usage is often subjective.

Tina Blue writes sagely about Comma usage with Compound Predicates:

As with other compound sentence elements, a compound predicate generally is not separated by a comma. Occasionally, however, if the parts of a compound predicate are unusually long, or if the writer feels the need for special emphasis, a comma can be used with a compound predicate. Such commas should be treated as a heavy spice, though, and used sparingly.

So the rule appears to be 'you must not use this construction unless you want to'.

My sort of rule.

I might use it here:

After dinner, the students will go into the lecture theatre and library and study to improve their knowledge of the great outdoors.

There's nothing wrong with the sentence but there are better ways to formulate it, especially for a book.

I would formulate it like this:

"She sighed before resuming/continuing to whisper."

When you formulate it this way you can add detail more easily.

Example:

"She sighed before resuming/continuing to whisper, unaware of her changed surroundings."

When you try it with the sentence you are using:

"She sighed, and began to whisper again, unaware of her changed surrounings."

It doesn't sound quite right does it?

Hope this helps :)

The only reason it could be considered wrong, is that in the first case the past perfect can be used, as it had happened before she started to whisper.

The word 'again' makes me do a double take. Did she sigh again or just once? She sighed again. She began whispering again. The comma makes me feel like the is doing two things... again.