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When molecules have resonance their e- exist in a hybrid of their lewis structures. Their e- can make those bonds (delocalized) to lower their repulsion. The result is a more stable atom. A molecule with more Lewis structures is more stable.
Another way to look at this (I think) is to look at bond order. If a molecule has resonance, its bond order will be higher than a molecule with only single bonds. In other words, its "average" bond strength will be in between that of a single bond and a double bond, and since multiple bonds are more stable than single bonds, the molecule with an "average" bond strength higher than a single bond will be more stable than a molecule that only has single bonds.
Since Resonance structures lie on the fact that a molecule is able to have two or more valid electron dot structures. therefore in calculating the average (hybrid) of the structures, it will be more stable compared to a single bond.
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