Hudhayfah ibn Al-Yaman (may Allah be pleased with him) narrated that the Prophet (peace and blessings be upon him) said, " Indeed whenever a believer meets another believer takes his hand and shakes it, sins fall from both of them the same way leaves fall off a tree."
Collected by At-Tabarani in his Awsot (245) graded Saheeh by Shaykh Al-Albani in his Silsilahtus Saheeh. 6/431
Benefits about this topic:
Shaykh Al-Albani 1421H (may Allah rest him in Jannah) wrote in Silsilahtus Saheeh; It is sunnah to take a person's hand and shake it. And to take both a person's hands and shake them is contradictory to the sunnah; like some of the mashaykh do.
Shaking hands is a legislated act at the time of parting as well. It is an innovation to shake hands directly after the prayer.(end of quote from Shaykh Nasir.[ 1/ 52] )
Al-bara' ibn 'Aazib (may Allah be pleased with him) said," Shaking hands is a complete greeting." [ # 968 Saheeh Adabul Mufrad]
The first people to introduce the handshake were the people from Yemen [ # 967 Saheeh Adabul Mufrad by Imam Al-Bukhari]
Translated and compiled by Abu Aaliyah Abdullah ibn Dwight Battle- aka- Adam due to the names of my boys; Muhammad, Ibrahim, Yaqub, Shuaib, Zakkariyah and Ayyub.