Any idea why they started using cast members to welcome the viewer to the show and then at the end of the show, tell them to watch next week in season five?

Is there a term for "stay tuned for next week's show" or "We hope we see you next week on"...?

It was probably a Don Fedderson thing. I remember Fred MacMurray doing it on M3S too.

I think they call it a bumper.

Yes, both Family Affair and My Three Sons used the "Welcome to our show" bit, usually along with a sponsor tag. Starting with season 10 of My Three Sons, they evidently filmed an alternate version with no sponsor tag, so it was used in syndication for awhile, until they evidently found they could save 5 seconds by not using it.

Original episodes of some sitcoms included a tease for next week's show. A few episodes of Newhart included them (probably accidentally) in syndication. Ditto for Father Knows Best, where probably every episode had Robert Young teasing the next episode, and the sponsor.

Notably, Home Improvement had some very clever animated bumpers, which made it into syndication for awhile, but not onto DVD sets. Bewitched had a clever bumper which didn't make it into syndication, but did make it onto later season DVD sets. And some, like Brady Bunch, were shown only on original network airings. Sitcoms primarily for kids, like Silver Spoons and Small Wonder, had bumpers before all ad breaks. I have a Silver Spoons set (home recorded off MeTV Chicago) which includes all the original bumpers with the season 4 episodes. Syndicated Everybody Loves Raymond had an opening bumper, but CBS did not use it on the original airings. Two And a Half Men has a short mid-episode bumper, included in syndicstion but was not on original airings.