I don't understand why combustion releases energy? So I've tried to wrap my head around fire. To keep it simple, I explain it with cellulose combustion. But to explain exotermic reactions, I show the example of methane combustion. Here is the detail in the bonds energy throughout the reaction:

What I don't get is that the energy to break the bonds is INFERIOR to the energy needed to create new bonds! How is that possible? With the activation energy? And so the surplus of activation is dissiped as heat?
Everyone says that:

"What happens is that breaking bonds between atoms releases energy, while bonding consumes energy. In a combustion reaction the bonds being broken contains more energy than the bonds formed, which means that the extra energy is being released into the surroundings as heat."

Well here we can clearly see that the bonds being broken contains LESS energy than the bonds formed?
Can someone enlighten me please?
 A: I think "bonding energy" or "energy required to create a bond" is a very unhelpful concept.  Although it is useful to accurately keep track of energy balances in chemical reactions, it is not a good description of what is "really" going on.
$\rm{CH}_4$ has more energy than $\rm{CO}_2$ in the same way that a pyramid balanced upside down on its tip has more energy than a pyramid lying on a flat face.  Let's say there is a thin rod bracing the pyramid while it is balanced on the tip, so it is "stable."  If a marble comes in and knocks over the rod, the pyramid will fall, and the rod and marble will get knocked across the table by the energy that is released by the pyramid falling.
Or again, let's envision a mouse trap, armed and ready to spring.  The spring is in an energized state, held in place by a little latch.  But it can't do anything unless that latch is moved.  Then if a marble falls on the trigger, the mouse trap will snap shut, and will jump up and bounce around and knock the marble across the room from the energy of the spring releasing.
Chemical reactions happen in the same way.  In these macroscopic examples, the energy to kick over the pyramid support, or the energy to trigger the latch, is the activation energy of the reaction.  The kinetic energy of the marbles and the mouse trap flying around (as well as the noise and heat produced) is the heat released by the reaction, and is equal to the difference in potential energy of the pyramid between the "propped up" and "laying flat" states, or the difference between the spring being tightened and relaxed.   And note that the kinetic energy released is greater than the activation energy required to set it off – this is the case for an exothermic reaction.
Likewise, in a combustion reaction, the molecules become reconfigured into a lower-potential-energy state, and that difference in energy is transferred into the kinetic energy of the molecules involved, which we see macroscopically as a rise in temperature and heat emission.  I should say "molecules and photons involved" because some of the energy can be released directly as light.
There is no "bond that is formed" in the sense that it eats up energy to bond a $\rm{C}$ atom to an $\rm{H}$ atom.  Rather, the two atoms become trapped together by the electric attraction between them – just like the Earth and Moon are "bound together" by gravity – and do not have enough energy to move apart.   It requires an energy input to move them apart.  This "hypothetical" or potential energy* that is not present, but would be needed in order to separate them, is referred to as the bonding energy.
*Note: I probably shouldn't call this "potential energy" in this case, since it is not – not in the sense that the pre-loaded mousetrap spring has potential energy.  This energy is the difference between the energy the molecules have when they are bound vs. when they are free.
A: Its a very poorly worded explanation/perspective ... you were unfortunate to find it. When a stick of dynamite goes off all scientists would say energy is released. The creation of stronger bonds releases tremendous EM energy in terms of photons and molecular vibrations ... i.e hot gases that expand rapidly and IR photons that spread heat. Closely within the explosion strong photons (UV , strong UV, extreme UV) rip apart molecules and a chain reaction is unstoppable.
A: For an exothermic reaction the total internal energy of the reactants is greater than the total internal energy of the products; the decrease in internal energy appears as increased kinetic energy of the products, sometimes called the energy released.
This is true for both chemical and nuclear reactions. Thanks to Einstein, the internal energy can be expressed as $E = mc^2$ so for an exothermic reaction (rest) mass, $m$, is converted to kinetic energy.  For an exothermic chemical reaction the energy released per reaction is very small compared to a nuclear reaction.  For example, nuclear fission releases about 200 Mev per reaction, while an exothermic chemical reaction typically releases a few tens of eVs per reaction.
Classical thermodynamics (before Einstein) considers the change in internal energy for a chemical reaction as the "energy of formation" or the "enthalpy of formation".  See one of the thermodynamics textbooks by Sonntag and Van Wylen.
In an exothermic reaction mass is converted to kinetic energy.  In an endothermic reaction kinetic energy is converted to mass.
Look up the Q value of a nuclear reaction on the web for more information.
Sometimes in an exothermic reaction (chemical or nuclear) the kinetic energy of the products must be significantly greater than zero; for example, nuclear fusion where the Coulombic repulsion of positively charged interacting nuclei must be overcome for the nuclei to have a nuclear fusion reaction. Similarly, in combustion, energy (heat) must be added. (For fission from a neutral neutron, the reaction can be achieved with a low kinetic energy neutron.)
