Well I've only been running the Season Two missions lately, but I'll try to answer best I can.
The Debriefing logs are effectively the Players proof that the character went through the mission and obtained X nuyen, Y Karma, and Z etc (special gear, faction, street cred and such) on said mission.
In addition, the GM is encouraged to look over all the players Debriefing logs for missions that tie into the one he's running, and use the outcomes to help tailor the mission. For example, if the players have all run a mission where they encountered a Mercenary Bodyguard and killed him, the GM is encouraged to rename that Bodyguard if he shows up again two missions later.