In this answer, I will point out an inadequacy of the reasoning based on resonance structures and I will also provided another MO perspective, which I believe is more convincing, on the issue.
A flaw in the resonance argument
As presented by Abcd, the phenomenon can be explained without use of MO theory and purely based on resonance structures. Please refer to his answer for the argument. Based on this argument, we may actually expect the $\ce {C=C}$ double bond to become weakened. Since if the two zwitterionic resonance structures were to be significant, the central double bond should naturally have less double bond character and more single bond character. However, data presented by Skancke & Boggs (1979) show that the $\ce {C-C}$ bond length is in fact shorter in the cis isomer ($\pu{1.311 Å}$) than in the trans isomer ($\pu{1.320 Å}$)$^1$. This suggests that the resonance stabilisation may not be the cause of this phenomena.
Another MO perspective
Fleming (2009) provides an explanation which I feel is more convincing. Essentially, the idea is that extra stabilisation in the cis form is derived from the interaction between the antiperiplanar $\ce {C-H}$ $\sigma$ and the low-lying $\ce{C-Cl}$ $\sigma^*$ MOs. Jan provides more details on this here when he explains the cause of a similar effect responsible for the gauche conformation being favoured in 2-fluoroethanol. Another phenomenon that Fleming touches on which he uses the same explanation for is the preference for the gauche conformation in 1,2-difluoroethane. It is also explained in the same paragraph on p. 89.
The same explanation can be used for 1,2-dichloroethene. This is because chlorine is also a rather electronegative atom, although less so than fluorine. Thus, we can still expect the $\ce {C-Cl}$ $\sigma^*$ MO to be low enough in energy to interact with the the $\ce {C-H}$ $\sigma$ MO.
References
- Skancke, A.; Boggs, J. E. Structure of 1,2-Difluorocyclopropane and the "Cis Effect". J. Am. Chem. Soc. 1979, 101 (15), 4063–4067.
doi:10.1021/ja00509a009
- Fleming, I. Molecular Orbitals and Organic Chemical Reactions. John Wiley & Sons, Ltd. United Kingdom, 2009.