I think I know the reason for that statement.
It is usually given when the GB member is going to talk about something that is potentially controversial or new to JWs. On such potentially controversial issues, there is the risk that the GB member would either be hated or loved for presenting it.
When it is a new or somewhat unconventional issue, there is also the potential for JWs to be wondering if the GB member is giving his own views or that of the organization.
The statement: "the Governing Body has asked me to..." serves to separate the potentially controversial or unconventional message from the lone GB presenter, putting it squarely unto the Governing Body/Faithful and Discreet Slave collective. That way, JWs are being asked to accept the bitter pill as coming from the God appointed Faithful and Disreet Slave who must be obeyed, rather than merely being the view coming from rubber-faced Lett, or Uptight Tony.
Any praise that a new, innovative or unconventional message brings must also go to the collective and not the lone GB presenter - so that the other GB members don't get jealous or suspicious about a lone GB member gaining more praise and prominence in the eyes of all JWs because of a message he happened to have presented.
It might also be a way of suggesting to JWs that what is about to be said is a very important message coming from the faithful and discreet slave and may thus indicate to us awake ones that this is the real "meat" of the program - the key bit of indoctrination that Watchtower wants to feed the JWs with for the month.
You see, the JW broadcast program is very similar to a study edition of the Watchtower. There are the experiences of JWs and other fluff pieces about the work of the organization. These mirror the supplementary articles for personal reading, found in the study edition. There there is the "the Governing Body has asked me to..." segment of the program which mirrors the congregation Watchtower study articles in the study edition.