Conjunctions should be used to conjunct two different parts (sentences, words, ...) so, technically speaking, starting a sentence with them is broadly considered a misuse.
But exactly for being a misuse, such construct has the effect of drawing extra attention on the sentence itself.
Using the form
E SENTENCE_B
as opposed to
SENTENCE_A e SENTENCE_B
immediately highlights the absence of a sentence joined before it and makes the reader wonder what the author omitted.
Such omission can be for instance the situation in which SENTENCE_B takes place, and we find many examples in poetry.
As an example [1], G. Pascoli starts his poetry Il gelsomino notturno with
E s'aprono i fiori notturni,
nell'ora che penso ai miei cari
The sentence after the E describes something happening at night, but there a whole lot of content unexpressed before it, namely the situation of waiting during the night, before the flowers disclose.
In this other example [1]
E, dopo tanti discorsi, finalmente l'accordo!
The conjunction E implies a phase of discussion before the deal. Using the E allows the writer not to explicitly mention it, but rather to suggest that the current situation is a consequences of previous events.
[1] Examples taken from http://www.accademiadellacrusca.it/it/lingua-italiana/consulenza-linguistica/domande-risposte/sulluso-congiunzioni-allinizio-frase