What is the reason why Qur’aan is recited out loud in some prayers and silently in others?

Firstly: 

Reciting out loud in the prayers in which the Prophet
(peace and blessings of Allaah be upon him) recited out loud, and silently
in the prayers in which he recited silently, is one of the Sunnahs of
prayer, not one of the obligatory duties. But it is better for the
worshipper not to go against the Sunnah of the Prophet (peace and
blessings of Allaah be upon him). 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

Reciting out loud in the prayers where this is done is not
obligatory, rather it is what is better. If a person recites silently in a
prayer in which it is prescribed to recite out loud, his prayer is not
invalid, because the Prophet (peace and blessings of Allaah be upon
him) said: “There is no prayer for one who does not recite the Essence of
the Book (al-Faatihah).” He did not specify whether this recitation is to be
out loud or silent. So if a person recites what he is required to recite,
silently or out loud, then he has fulfilled his duty. But it is better to
recite out loud in the prayers where doing so is customary, as in Fajr and
Jumu’ah prayers. 

If a person deliberately does not recite out loud when he is
leading the prayer, his prayer is valid but it is lacking.

 If a person is praying alone, he has the choice between
reciting out loud or silently. He should look at what is more likely to
increase his focus and humility in prayer, and do that. 

End quote. 

Majmoo’ Fataawa Ibn ‘Uthaymeen,
13/73 

Secondly: 

The basic principle is that a Muslim is obliged to adhere to
the laws of Allaah, without making that dependent on whether he knows the
reason or wisdom behind it. That does not mean that he should not try to
find out the wisdom behind it, after following the command and adhering to
the guidance. 

See question no. 20785 and
26862. 

Thirdly: 

The scholars of the Standing Committee were asked: 

Why do we pray Zuhr and ‘Asr silently and Maghrib and ‘Isha’
out loud? 

They replied: 

We do that following the example of the Prophet (peace
and blessings of Allaah be upon him). We recite silently in the prayers in
which he recited silently, and we recite out loud in the prayers in which he
recited out loud, because Allaah says (interpretation of the meaning): 

“Indeed in the Messenger
of Allaah (Muhammad) you have a good example to follow”

[al-Ahzaab 33:21] 

And the Prophet (peace and blessings of Allaah be upon
him) said: “Pray as you have seen me praying.” Narrated by al-Bukhaari in
his Saheeh.

 Fataawa al-Lajnah al-Daa’imah,
6/394, 395 

Shaykh ‘Abd al-‘Azeez ibn Baaz was asked:

 Why is it prescribed to recite out loud in Maghrib, ‘Isha’
and Fajr, and not in the other obligatory prayers? What is the evidence for
that? 

He replied: 

Allaah knows best the reason why it is prescribed to recite
out loud in those prayers. The most likely reason – and Allaah knows best –
is that at night and at the time of Fajr prayer, people are more likely to
benefit from recitation that is done out loud, as there are less
distractions around them than at the time of Zuhr and ‘Asr. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz
(11/122) 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have
mercy on him) was asked: 

What is the reason for reciting out loud in Jumu’ah prayer? 

He replied:  

The reason why it is recited out loud is – and Allaah knows
best – firstly so as to unite the people behind one imam, because if they
are united behind one imam and listening attentively to him, that is a
better form of unity than if each one of them were to recite to himself. In
order to achieve this the people must all be gathered in one place, except
in cases of necessity. 

The second reason is that the imam’s recitation out loud
during the prayer is a completion of the two khutbahs. Hence the Prophet
(peace and blessings of Allaah be upon him) used to recite in Jumu’ah
prayer soorahs that were appropriate, either al-Jumu’ah and al-Munaafiqeen,
because the former mentions Jumu’ah and encourages us to attend the prayer,
and the latter describes hypocrisy and denounces its people; or he would
recite al-A’la and al-Ghaashiyah, because the former mentions the beginning
of creation and describes various creatures and the beginning of divine
laws, and the latter describes the Resurrection and rewards and
punishments. 

The third reason is so as to distinguish between Zuhr and
Jumu’ah. 

The fourth reason is so that it will be similar to Eid
prayer, because Jumu’ah is the weekly “Eid”. End quote. 

Majmoo’ Fataawa Ibn ‘Uthaymeen,
16/112 

See also the answer to question no.
65877. 

And Allaah knows best.