There is an activation energy because forming the transition state is (usually) energetically unfavorable. It's like trying to roll a ball over a hill. If you don't push it hard enough, it's just going to roll right back down in your direction. The activation energy is the difference in free energy (enthalpy/entropy) between the reactants and the transition state; it is related to the amount of energy required to break required bonds, but I don't think that's necessarily the best way to look at it - although for general chemistry or high school chemistry level, it's sufficient.
Plus - and I can't stress this enough - these energies we speak of represent statistical averages. Heat isn't really something that's absorbed and emitted by single molecules that are reacting. It's a representation of the average amount of kinetic energy in a system. When heat is released from an exothermic reaction, it's because the average potential energy of the molecules in the system after the reaction is less than that of molecules before the reaction, and the difference has been converted into kinetic energy (vibrations, rotations, translations). It's not because each reacted molecule gives out a little squirt of "heat". In other words, energy is more a state of the system than a true "reactant" or "product". At least, that's the way I like to view it.
As for the decomposition reaction, shouldn't there still be an activation energy since I have to break those intramolecular bonds?
Yes, there usually is. Sometimes it's so small that it takes almost nothing to overcome (as in shock sensitive explosives like nitrogen triiodide or fulminated mercury), and in these cases the "transition state" is difficult to precisely define. Also, I'll remind you that activation energy is NOT just enthalpy. It's entropy as well. A lot of times a good portion of the energy required to make a reaction go is not just a matter of breaking and forming bonds. It's the entropic change required for arranging a very specific structural transition state form. If the transition state of a decomposition requires entropically unfavorable rearrangement of solvent molecules, for example, this could contribute to a reasonably large activation energy even if the amount of energy required to break bonds and interactions is relatively small. At least in principle; can't think of a good example off the top of my head.
Just keep in mind that EVERY chemical will decompose if enough heat energy is put into the system.