What's the explanation for the chelate effect?
The chelate effect is generally agreed to be a thermodynamic effect caused by the change in entropy upon binding of a bidentate ligand. 



General description of the chelate effect:

Consider a coordination complex,  [M]L2, where L = a standard monodentate ligand. If we now add a bidentate ligand, two monodentate ligands are released. 


The ligand substitution reaction in the forward direction is therefore generally favourable:  ∆S, the entropy term is positive due to two molecules reacting to become three, with this favourable entropy term having the effect of tending to make ∆G negative (negative ∆G is favourable, i.e. the reaction likely to proceed).[*] 

Once a bidentate ligand is bound, it then becomes less favourable to go in the other direction (the entropy term would be negative due to entropy decreasing in the system, which in turn would tend to make ∆G positive, and hence the reaction unfavourable). 

Example and thermodynamic data: 

To give a more concrete example demonstrating the significance of the entropy term, consider the following reaction and associated thermodynamic data (taken from Physical Inorganic Chemistry, S. F. A. Kettle, 1996): 





Clearly, the entropy term dominates, causing ∆G to be negative in both cases (hence causing the forward reaction to be favoured in both cases).

[*]: The caveat here is that we're assuming that enthalpy really isn't contributing a whole lot, this is clearly a vast over-simplification, and there are cases where enthalpy becomes important (i.e. when the complex is significantly more stable than the starting materials). We're also ignoring the fact that the act of binding a bidentate ligand itself has some unfavourable entropy, since we're containing the ligand and removing some conformational flexibility