Why is it common for cheap bikes to have the frame sawn in half below the saddle? These days cheap bikes often seem to come with the frame sawn in half below the saddle. 

What does the manufacturer gain from this design, and why is it associated mostly with cheap, low quality bikes? 

What disadvantages are there to the bike rider? I assume that it's not a great feature since quality bike makers don't seem to opt for this design.
The seat tube is split because it has rear suspension - the rear of the bike rotates around a pivot located just in front of the bottom bracket.  Several early full-suspension bikes used a similar design for the rear pivot; this design was then knocked off by the lower-end manufacturers looking to cash in on the popularity of mountain bikes.  It's extremely simple and gives the bike the expensive "full-suspension" look.  Unfortunately it mostly just adds weight, pedal bob, and complexity to an already heavy turd of a bike.

Most modern full-suspension bikes use more expensive rear linkages and advanced suspension designs, so this older design has almost entirely been relegated to department store bikes.