In the name of Allah, the most compassionate, the most merciful
According to the holy Qur'an:
وَ إِذا قُرِئَ الْقُرْآنُ فَاسْتَمِعُوا لَهُ وَ أَنْصِتُوا
So when the Qur'an is recited, then listen to it and pay attention
that you may receive mercy. (7:204)
One of the points which we ought to notice when the Quran is recited, is being silent in order to listen to its verses to meditate in them. The apparent of the mentioned verse shows that it is a general ruling, but in accordance with the narrations and likewise the unison of the Olama' (or scholars) it could be presumption that generally it is a Mustahab (recommended) ruling and it is worthy and suitable for people not to speak during the Qur'an is recited.
Since the holy Qur'an is not solely a reciting book, haply it is an understanding book which Muslims must perform it in their life.
The only item which make us be silent (which is compulsory) is when the Qaari is Imam-Jamaa' (in truth, when saying prayers).
According to Tafsir-Nemuneh:
The reason of the order of being silent (when the Quran is recited in the
mentioned case) is that humans pay attention to the verses of the holy
(Tafsir-Al-Nemuneh, Vol7, Pg71)