Why is it common for cheap bikes to have the frame sawn in half below the saddle?
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.