What is the reason that recitation (qira’ah) is done aloud in some prayers and silently in others?

Acts of worship are tawqīfī; that is, it is not possible to know the reasons for their obligation or the manner of their performance through reason alone. They are performed as Allah has commanded and as the Prophet (peace be upon him) practiced them. Prayer is no exception. The Prophet (peace be upon him) said, “Pray as you have seen me pray.” (Bukhari, Adhan, 18 [631])

For this reason, the practice of reciting silently in obligatory prayers performed during the day and aloud in those performed at night is based on the Sunnah of the Prophet (peace be upon him) and the practice of the noble Companions who learned the religion from him and taught it to others. (See Bukhari, Adhan, 96-105 [759-774]; Muslim, Salat, 47-48, 149-179, 201 [398, 449-465, 475])

Source: T.C. Diyanet İşleri Başkanlığı, Din İşleri Yüksek Kurulu