Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) says that when this verse was revealed, the Messenger of Allah (peace and blessings be upon him) at that time used to worship in hiding in Makkah. When he prayed, he would recite aloud. When the polytheists heard the Qur'an, they reviled the Qur'an, and also reviled the One who revealed it and the one who brought it. So Allah Almighty said to His Prophet (peace and blessings be upon him): "And do not be loud in your prayer, nor silent in it" [Surah al-Isra': 110], meaning: O Prophet, neither recite aloud in your prayer nor recite completely quietly; that is, recite with enough voice to make only your companions hear. "And seek a way between that" [Surah al-Isra': 110], that is, recite with enough voice that your companions may hear the Qur'an and memorise it.
Imam Muslim (may Allah have mercy on him) also narrated this hadith in his Sahih with his own chain. At its end it says that Allah Almighty said to His Prophet (peace and blessings be upon him): "And do not be loud in your prayer" [Surah al-Isra': 110], so that the polytheists do not hear your recitation; "nor silent in it" [Surah al-Isra': 110], too quietly; rather, make your companions hear the Qur'an; "and seek a way between that" [Surah al-Isra': 110], that is, recite with enough voice that your Companions may hear it from you and memorise it.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الصلاة / 3070