Hadith: We used to speak during prayer; a man would talk with his companion at his side in the prayer until [this verse] was revealed: {...And stand before Allah in devout obedience.} So we were commanded to keep silent, such that we were forbidden to talk

عن زَيْد بن أَرْقَمَ رضي الله عنه قال: «كُنّا نتكلم في الصلاة، يُكَلِّمُ الرجل صاحبه، وهو إلى جنبه في الصلاة، حتى نزلت ((وقوموا لله قانتين))؛ فَأُمِرْنَا بالسكوت ونُهِينَا عن الكلام».
[صحيح] - [متفق عليه]

Zayd ibn Arqam (may Allah be pleased with him) reported: We used to speak during the prayer; a man would talk with his companion at his side in the prayer until [this verse] was revealed: {...And stand before Allah in devout obedience} [Sūrat al-Baqarah: 238]. So we were commanded to keep silent, such that we were forbidden to talk [to each other during prayer].
[Authentic hadith] - [Narrated by Bukhari & Muslim]

Explanation

The ritual prayer is a communication between the servant and Allah. So in prayer, one should not allow anything to distract him from this private communication with Allah. Zayd ibn Arqam (may Allah be well pleased with him) says that the Muslims, at the beginning of Islam used to talk to each other during ritual prayers as much as they needed. So one would talk to his companion beside him during the prayer. The Prophet knew of that, but he did not disapprove of it. Since one converses privately with Allah during prayer, no contact should be made with humans. Hence, Allah, the Exalted, commanded the Muslims to observe their prayer with silence, saying: {Maintain with care the [obligatory] prayers and [in particular] the middle prayer and stand before Allah, devoutly obedient} [Sūrat al-Baqarah: 238]. So the Companions came to know from this verse that talking during the prayer is forbidden, and they refrained from talking from that moment on.