Imam An-Nawawi wrote in his Sharh Sahih Muslim [Arabic text and English translation below]:
As for his saying, peace be upon him,:
In it is a tremendous encouragement to spread peace and to offer it to all of the Muslims, those you know and those you do not know, such as had come in other Hadith. And the Salaam [greeting of peace] is the first of the causes of reconciliation, the key to attracting affection – this is the significance of the Salaam – and in spreading it is the possibility ofأَفْشُوا السَّلَامَ بَيْنَكُمْSpread peace between yourselves
