Firstly, it isn’t a rule. The next time you are challenged by anyone who fetishizes this issue, ask for their opinion of the King James Bible, frequently cited as one of the greatest repositories of well-turned English phrases. The fragment “And it came to pass” occurs 396 times. The vast majority of these are at the beginning of sentences.
Secondly, pedantically, it isn’t grammar. That may surprise you, but think of it this way: grammar is about the structure of language – arranging the nouns, verbs and other components so that they make sense. Sentences – in the sense of units that start with capital letters and end in full stops – are not part of grammar; they are elements of orthography. Try saying the following passage aloud: “There are seventeen pirates on this ship, every pirate has a cutlass and the most dangerous pirate is One-eyed Jake.” Now ask anyone who is willing to listen to you reciting such nonsense how many sentences there were in that speech. The answer could be one, two or three. From a grammatical viewpoint, it doesn’t matter. Generally, you can’t hear whether the speaker put in a comma or a full stop after “ship”. On the other hand, from a point of view of orthography – the presentation of the written form of the language – the punctuation and capitalisation matters: it helps to clarify meaning and improves readability.
The third point about whether or not sentences can begin with ‘and’ is that putting it that way mis-states the intention of the guideline. ‘And’ is a conjunction. It says that whatever comes after the word is joined to whatever went before. Go back to the King James Bible. Those sentences beginning “And it came to pass” are using ‘And’ to indicate that what follows is part of the same story. That’s story, not sentence. We’re definitely onto a new subject (because it came to pass that something new happened with a new main verb) and therefore onto a new sentence, but it is a continuation of the same story. Or take the following fragment of horror from Shakespeare’s King John:-
Arthur: Must you with hot Irons burne out both mine eyes?
Hubert: Young Boy, I must.
Arthur: And will you?
Hubert: And I will.
In this case, ‘And’ indicates continuity with the previous statement, but since the previous statement was made by another speaker, it is punctuated as the start of a new sentence.
A case can also be made for emphatic use: the writer may wish to give prominence to a sub-clause. And this demonstrates the appropriate stress. Similarly, particularly in recorded speech, it may demonstrate a linked afterthought.
So if the rule doesn’t apply here, where does it apply? This particular rant was occasioned by a play script that contained 85 sentences starting with the word ‘and’. Many of them were legitimate, for the reasons cited above. Others were not, because the structure reduced rather than enhanced the meaning. Take the following:
But let us indeed continue. And trust that a more seemly life emerges.
The second sentence does not have a subject, or rather the subject is in the first sentence, and the second sentence only makes sense as a continuation of the first. As a general test of this, do away with the ‘and’ and see if the sentence still makes sense. The examples from Kings James and John pass that test. The above example does not.
The point of the rule is not that you shouldn’t start a sentence with ‘and’; the point of the rule is that you should not put a full stop in the middle of a sentence.