Why do so many TV shows have a step down from the front door into the living room?

Sunken living rooms were very popular in houses built in the 1960s and 70s, where they are often used in open plan designs to create "rooms" without walls, or to add ceiling height without raising the roof level.

They define the space. Sunken spaces are great for large, open concept homes. When walls are few, having rooms on a lower level clearly delineates the space without destroying any sight lines. The feeling of openness is preserved without losing any definition between, say, the living room and the kitchen.

freshome.com

So on shows made in or set during the 1970s (or in homes that were built during the 1970s), you would expect to see sunken living areas, as a reflection of the architectural trends of the times.

The Dick Van Dyke Show has also been identified as both a reflection of and contributor to that trend.

Hollywood caught on, making a sunken living room the focal point of home life on the The Dick Van Dyke Show in the ’60s. The trend had been validated.

Soon homeowners across the world were scrambling to be en vogue with step-down living spaces.

houzz.com

Given how influential The Dick Van Dyke Show (and later The Mary Tyler Moore Show) was, you would also expect later shows to have been influenced by it's sets.

There is also the obvious practical benefit for a multiple-camera sitcom that is demonstrated in your image: raising the upstage area means that the cameras can shoot characters back there without the foreground set elements blocking them.