"It started to get dark."

Translation: A început să se însereze.

[DocPummello]

Incepuse este corect ca si a inceput

[Ray301962]

Why is t "A" used at the beginning?

[WarsawWill]

It's an auxiliary verb, the equivalent of "ha" in Italian, and "a" in French (see examples below). As in spoken French and Italian, the compound form of the Perfect is being used to express the simple past. This is gramatically similar to the Present Perfect in English (which, incidentally, is generally accepted in translations the other way):

(eu) am început - I have started
(tu) ai început - you have started
a început - it has started
etc

Its function will perhaps be clearer if we add an explicit subject:

Mihai a început să meargă la școală
- Mihai has started going to school

Talking about atmospheric conditions, the weather, the time, etc, in Romance languages the verb is generally used impersonally, without a subject, the exception being French. In French and English, we need an explicit subject. Compare:

Impersonal - subject pronoun not used:
A început să se sereze
Ha iniziato a farsi buio (it)

Explicit subject needed:
Il a commencé à devenir sombre (fr)
It (has) started to get dark

More on Romanian verb forms and their use here:
https://en.m.wikipedia.org/wiki/Romanian_verbs