Ask Mr. Smarty Plants

There is an excellent article of the basics with photographs from Wayne's Word, Palomar College in San Marcos, California about carnivorous plants, including the Dionaea muscipula (Venus flytrap). 

I will summarize the process as explained on that page and from other articles about the process:

The edge of the capture leaf has three sensitive hairs on the inside surface of the trap (the upper side of the leaf).  When an insect lands on the leaf and touches two or more of those sensitive hairs, or touches one of the hairs more than once, this precipitates a change in turgor pressure in the cells on the upper side of the leaf.  Since the cells lose liquid, this causes the leaf to fold over trapping the insect inside the leaf.  The stiff bristle-like hairs along both edges of the leaf interlock so that the insect cannot escape.  Once inside, glands on the inner surface of the closed trap release an enzyme that digests the insect and releases the nitrogen that the plant requires.  There is also a chemical aspect to the closing of the trap.  It apparently requires ATP (adenosine triphosphate), a nucleotide that serves as a biochemical energy source, to change the turgor pressure and close the trap.

It is agreed that the surface of the leaf changes shape.  When open, the upper surface of the leaf (where the sensitive hairs are) is convex.  When the trap closes that upper surface is concave.  The closing occurs rapidly—in 0.3 seconds (according to Alexander G. Volkov et al.  2008.  "Kinetics and Mechanism of Dionaea muscipula Trap Closing."  Plant Physiology 146:694-702).  The change in shape occurs because the cells on the upper surface lose fluid rapidly to the cells on the lower surface causing the upper surface to become concave and the lower surface to become convex, closing the trap.  As the insect struggles inside the trap, the sensitive hairs are further stimulated and the upper surface of the leaf loses more fluid and become more concave, thus further closing the trap. Charles Darwin had observed this and done experiments with the Venus' flytrap.  You can read Darwin's description of the Venus' flytrap online in Darwin, CR. 1875. Insectivorous Plants. London: John Murray. pp. 286-320.

The underlying mechanism of how the trap closes is still controversial, however.  Stephen Williams and Alan Bennett (1982. "Leaf Closure in the Venus Flytrap: An Acid Growth Response" Science 218 (4577): 1120-1122) suggested that a rapid lowering of pH (becoming more acid) caused the cell walls to loosen and change the turgor pressure between surfaces.  Others (DeGreef, see below) suggest it is an electrochemical mechanism responsible for the closure and Volkov et. al (see above) describes it as a "hydroelastic curvature mechanism".

You might like to read "How Venus' Flytraps Catch Spiders and Ants, Pt. 1"  in the Carnivorous Plants Newsletter 9 (3):65,75-78. 1980, and "How Venus' Flytraps Catch Spiders and Ants, Pt. 2" Carnivorous Plants Newsletter 9 (4):91,100. 1980 by Stephen E. Williams.

Also, "The Electrochemical Mechanism of Trap Closure in Dionaea muscipula" Carnivorous Plants Newsletter 17 (3):80-83,91-94. 1988 and "The Electrochemical Mechanism of Trap Closure in Dionaea muscipula. Addendum" Carnivorous Plants Newsletter 17 (4):106. 1988 by John D. Degreef.

Whatever the underlying mechanism is, it is truly an amazing plant!