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The description in terms of rectangles was noted by Sridhar Ramesh when he solved this.
All examples in this Bongard Problem feature arced line segments connected at endpoints; these segments do not cross across one another and they are nowhere vertical; they never double back over themselves in the horizontal direction.
Furthermore, in each example, there is a single leftmost point and a single rightmost point, and every segment is part of a path bridging between them. So, there is a topmost total path of segments and bottommost total chain of segments.
Any picture on the left can be turned into a subdivided rectangle by the process of expanding points into vertical lines.
Here is another answer:
"Right examples: some junction point has a single line coming out from either the left or right side."
If there is some junction point with only a single line coming out from a particular side, the point cannot be expanded into a vertical segment with two horizontal segments bookending its top and bottom (as it would be if this were a subdivision of a rectangle).
And this was the original, more convoluted idea of the author:
"Start with a string along the top path. Sweep it down, region-by-region, until it lies along the bottom path. The string may only enter a region when it fully covers that region's top edge and likewise it must exit by fully covering the bottom edge. Only in left images can this process be done so that no segment of the string ever hesitates."
Quite convoluted when spelled out in detail, but not terribly complicated to imagine visually. (See the keyword unwordable.)
The string-sweeping answer is the same as the rectangle answer because a rectangle represents the animation of a string throughout an interval of time. (A horizontal cross-section of the rectangle represents the string, and the vertical position is time.) Distorting the rectangle into a new shape is the same as animating a string sweeping across that new shape.
In particular, shrinking vertical lines of a rectangle into points means just those points of the string stay still as the string sweeps down.
The principle that horizontal lines subdividing the original rectangle become the segments in the final picture corresponds to the idea that the string must enter or exit a single region all at once. | 
