"Never mistake motion for action." - Ernest Hemingway

Soon I hope to have a tutorial document on this subject illustrated with animation. However, in the mean time you can read about the topic in an early survey paper I wrote or play with the interactive Java applets found on some of the links listed below.
• Godfried T. Toussaint, "Movable separability of sets," in Computational Geometry, Ed., G. T. Toussaint, North-Holland, 1985, pp. 335-375. This paper is a tutorial survey of research results obtained before 1985 related to disassembly and interlocking puzzles in two and three dimensions. This area is sometimes called local motion planning.
You will also find more in-depth material in my papers on the following more specific topics:
The research described in the above papers is in fact concerned with the general theory of interlocking polygons in the plane and polyhedra in space. Therefore the analysis and design of interlocking puzzles such as burrs is closely related to local motion planning as well as assembly and disassembly in manufacturing. The literature on burr-puzzles is very helpful for thinking about and visualizing these kinds of geometric problems.

• The Sofa Problem: The sofa problem asks for the largest (maximum-area) shape that can be moved around a corner in a corridor. This is still an open problem but upper and lower bounds on the solution exist. See the nice interactive applet where you can move several sofas of different lengths.
• Separability, and mobility in general, of objects are of course intimately related to immobility which, in turn, is the central issue of grasping and fixturing problems in robotics and manufacturing. Check out the postscript document below.
• Godfried T. Toussaint, "All convex polyhedra can be clamped with parallel jaw grippers," Computational Geometry: Theory and Applications, vol. 6, 1996, pp. 291-302. (with P. Bose and D. Bremner)