Some of the evidence for benzene having the structure detailed in other writeups (with a delocalised ring of electrons at the centre) as opposed to the Kekulé structure includes thermodynamic considerations.

Let's look at the enthalpy change for the complete hydrogenation of cyclohexatriene (essentially the Kekulée structure) to cyclohexane. The problem is, cyclohexatriene doesn't exist due to its instability. So we use our imaginations, and think as follows:
The enthalpy change for the hydrogenation of cyclohexene (like cyclohexane but with a double bond replacing one of the single bonds) is -120 kJ/mol.
The enthalpy change for the hydrogenation of cyclohex-1,3-diene (like cyclohexane but with two double bonds) is -233 kJ/mol.
This is roughly twice the amount for the same reaction for cyclohexene; we could therefore say that the enthalpy of hydrogenation of cyclohexatriene will be roughly three times that for cyclohexene: -360 kJ/mol.

But when real benzene is hydrogenated, the experimental value for the enthalpy change is -208 kJ/mol. Hence, less energy is given out when 1 mole of benzene is hydrogenated than when 1 mole of cyclohexatriene is hydrogenated. This means that benzene is more stable than cyclohexatriene.

That is the thermodynamic evidence for the structure of benzene. However, there are more hints that the delocalised-ring structure is the true picture, and some of these come from the reactions that benzene undergoes. If benzene was as Kekulé proposed, it would have discrete double bonds, and so would undergo electrophilic addition reactions. This is not the case. Instead, benzene undergoes electrophilic substitution reactions, at the end of which the delocalised ring of electrons is unchanged (although it may be disrupted during the reaction, for instance in the halogenation of benzene). This kind of reaction shows us that benzene must have this structure.