Praise be to Allah.
It was decreed that recitation in Zuhr and ‘Asr should be done silently because the time of these prayers is during the day, and at that time a person's mind may be preoccupied with a lot of thoughts, as he is busy with his work or trade or manufacturing. If the recitation were to be done out loud, his mind would be distracted and he would not listen attentively to the imam’s recitation; he would not listen to it or devote his attention to it; rather other things would come to mind and distract him. So he was commanded to recite to himself so that his recitation will make him think and ponder. It is well known that if people pray in congregation for Zuhr or ‘Asr prayer, they should not recite out loud, because then they would distract one another. So both the imam and the one who is praying behind him are enjoined to recite silently. But with regard to the prayers that are offered during the night, in most cases people are free from distractions so the imam is enjoined to recite out loud, so that others may benefit from his recitation.
With regard to the verse which you mention, it applies only to qiyaam al-layl and tahajjud. If a person is praying and there are people around him who are listening to his recitation and others who are asleep, then he should not recite so loudly as not to disturb those who are sleeping or so quietly as to cause those who are listening to him to miss out on any benefit; rather he should recite at a moderate volume, enough to be heard but not so loud as to disturb those who are asleep. It also states in the verse that the Messenger (blessings and peace of Allah be upon him) used to recite out loud and the mushrikeen heard him and reviled the Qur'aan and the one who had bought it, so he was forbidden to recite out loud and he began to recite silently, but then the members of his household and those who were listening to him missed out on the benefit, so he was enjoined to recite at a moderate volume.