Remember that if a reaction is exothermic:
ΔH = energy used in bond breaking - energy released in bond making
and
energy used in bond breaking < energy released in bond making
Here's a nice way to think of it qualitatively (though not necessarily a fully accurate one).
Since bonds have low potential energy (and therefore are stable), it requires an input of energy (equal to the activation energy) to make that bond unstable and break. For bond making, the products must achieve some stability (otherwise they won't form) and therefore their potential energy must go from high to low. So, they must relinquish some excess potential enegy into the surroundings (e.g. convert it to heat) to lower their potential energy.
Bond breaking:
For simplicity assume we have a strong acid like HCl. Due to the highly electronegative nature of Cl, it will tend to suck out electrons from the H to give H+ and Cl- in solution. This means that HCl is generally "unstable" in solution and the bond breaking happens very readily and therefore very little input energy is required.
Bond making:
If you look at the Lewis electron dot structure of H2O, there are lone pairs around the oxygen atom of the molecule and also it is highly electronegative so it contains a partial negative charge. The H+ from the HCl is positively charged and is therefore attracted to the lone pairs and the partial negative charge of the oxygen component of the H2O to form H3O+.
Since this compound is quite stable due to electrostatic attraction, then the potential energy of the hydronium ion must be quite low and therefore a large amount of excess energy is released in order to pull the energy levels from the high activation peak to a lower level.
Putting this altogether we have low energy absorbed to break bonds but high energy released to make bonds. The net effect is more energy released and thus it is exothermic.