What is the reason why Quran 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 Allah 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 Allah be upon him). 

Shaykh Ibn ‘Uthaymeen (may Allah 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 Allah be upon him) said: “There is no prayer for one who does not recite the Essence of the Book (al-Fatihah).” 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’ Fatawa Ibn ‘Uthaymeen, 13/73 

Secondly: 

The basic principle is that a Muslim is obliged to adhere to the laws of Allah, 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 Allah 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 Allah says (interpretation of the meaning): 

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

[al-Ahzaab 33:21]

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

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

Shaykh ‘Abd al-‘Azeez ibn Baz 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: 

Allah knows best the reason why it is prescribed to recite out loud in those prayers. The most likely reason – and Allah 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’ Fatawa al-Shaykh Ibn Baz (11/122) 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah 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 Allah 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 Allah 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’ Fatawa Ibn ‘Uthaymeen, 16/112 

See also the answer to question no. 65877. 

And Allah knows best.