The reason why Quran is recited out loud in the nighttime prayers and silently in the daytime prayers
Question

What is the reason why Quran is recited out loud in Maghrib and ‘Isha’ prayers? 
What is the reason why it is recited aloud only in the first two rak’ahs of ‘Isha’ and not in all four rak’ahs?.
Praise be to Allah.Firstly: 
The Muslim is required to follow the Prophet (peace and blessings of Allah be upon him) and follow his example, whether he knows the reason behind a ruling or he does not. Allah says (interpretation of the meaning): 
“Indeed in the Messenger of Allah (Muhammad) you have a good example to follow for him who hopes for (the Meeting with) Allah and the Last Day, and remembers Allah much”
[al-Ahzaab 33:21]
He should also be certain that Islam is based on great wisdom, but this wisdom may be known to us or we may be unaware of it, or we may know some of it and not the rest of it. 
There is nothing wrong with a person asking the reason and trying to find out, because knowing it will increase his knowledge and peace of mind. 
Secondly: 
Some of the scholars have pondered the reason why Quran is recited out loud in the nighttime prayers and silently in the daytime prayers, and the summary of what they have mentioned concerning that is: 
The night is a time when people are quiet, alone and more focused, so it is prescribed to recite Quran out loud at that time so as to make manifest the pleasure of a person’s conversing with his Lord, and so that the heart, tongue and ears will be involved in the recitation. 
This meaning is suggested by the verse in which Allah says (interpretation of the meaning): 
“Verily, the rising by night (for Tahajjud prayer) is very hard and most potent and good for governing oneself, and most suitable for (understanding) the Word (of Allah).
7. Verily, there is for you by day prolonged occupation with ordinary duties” [al-Muzzammil 74:6-7]
Ibn Katheer said: 
The point is that this rising may be at any time of night. What is meant here is that when praying at night, both the heart and the tongue are focused on the recitation, hence Allah says (interpretation of the meaning): “is very hard and most potent and good for governing oneself, and most suitable for (understanding) the Word (of Allah)” i.e., the mind will be more focused when reciting and will understand the words more than during the day, which is a time when people move about and there is a great deal of noise, and it is the time of seeking provision. End quote. 
Al-Qurtubi (19/40) said: 
What is meant is that the heart, sight, hearing and tongue work together in harmony when reciting Quran, because voices and movements are stilled. End quote. 
Al-Sa’di (may Allah have mercy on him) said (p. 1058): “Verily, the rising by night (for Tahajjud prayer) is very hard and most potent and good for governing oneself, and most suitable for (understanding) the Word (of Allah)” i.e., it is more likely to achieve the purpose of the Quran, where both heart and tongue are focused, because there are fewer distractions, one can understand what one recites, and does the prayer properly. 
This is unlike the day, when these purposes cannot be achieved. Hence Allah says “Verily, there is for you by day prolonged occupation with ordinary duties” – i.e., going about to attend to your needs and living, which means that the heart is distracted and cannot focus fully. End quote.  
It says in Tuhfat al-Muhtaaj: 
There remains the wisdom behind reciting out loud – what is it? It may be that it is because night is the time when one is alone and people get together for lighthearted talk. So it is prescribed to recite Quran out loud at that time so as to make manifest the pleasure of a person’s conversing with his Lord. This applies only to the first two rak’ahs because the worshipper has more energy then. Because the day is the time of distractions and mixing with people, it is required to recite silently, because the daytime is not the best time to focus fully on conversing with Allah. The morning (Fajr) prayer is joined to the night prayers because its time is not usually a time when there are distractions, like Friday. End quote. 
Ibn al-Qayyim (may Allah have mercy on him) said in I’laam al-Muwaqqi’een (2/91): 
With regard to the distinction between the nighttime prayers and the daytime prayers, in terms of reciting out loud and reciting silently, there is great wisdom in that, because the night is a time when voices and movements are stilled, and hearts are focused and have renewed energy. The day is the time when people are busy with their day-to-day occupations, and the night is the time when a person can focus with his heart on what he is saying with his tongue. Hence the Sunnah is to make the recitations in Fajr longer than in other prayers. The Messenger of Allah (peace and blessings of Allah be upon him) used to recite between sixty and one hundred verses in them, and Abu Bakr used to recite al-Baqarah in them, and ‘Umar used to recite al-Nahl, Hood, Bani Israa’eel, Yoonus and similar soorahs, because the heart is more free of distractions when it has just been wakened from sleep. If the first thing that a person hears is the word of Allah in which there is all goodness, then these words will come to his heart which is free form any distraction, and will thus have a great impact on him. As for the day, the opposite is the case, so Quran is recited silently unless there is a reason, such as the large gatherings on Eid and Jumu’ah (Friday), and when offering prayers for rain and the eclipse prayer. On those occasions it is better to recite out loud and that is more likely to achieve the objective and is more beneficial, and because this serves to convey the words of Allah when there is a large gathering. End quote. 
May Allah increase us and you in beneficial knowledge and righteous deeds. 
And Allah knows best.