What does εἰς τὸ ἀκοῦσαι mean in James 1:19. Please, explain how the article works here?
Here is my overly literal translation of James 1:19 -
Know [this], my beloved brothers: However, let every person be quick unto the hearing, slow unto the speaking, and slow unto anger
In each case, I have translated εἰς as "unto" a simple preposition. In more idiomatic English, this might be translated as:
Know this my beloved brothers, let everyone be quick to listen, slow to speak and slow to anger.
Thus, εἰς τὸ ἀκοῦσαι is simply (literally) "quick to the hearing", or better, "quick to listen".
Dottard already gave the best answer, but to be technical "how the article works here" is really just a small addition to Dottards answer.
εἰς is a preposition.
ἀκοῦσαι is a verb form (a nominal form of the verb).
Prepositions normally stand before a noun phrase and govern its case.
The article turns the infinitive into a "noun" (nomen) and the preposition εἰς governs the accusative.
So it works as a "function word" functioning to turn a verb form into a noun. Greek can be very productive with this mechanism of creating noun phrases with an article.
The εἰς τὸ + infinitive is often used to indicate a purpose (in order to) or a result (so that).
"quick so that you hear/listen" = quick to listen
Dottard expressed that part better.