Electrons around the nuclei of atoms exist in various energy states due to quantum mechanical principles. Due to these principles, they are confined to certain configurations (shells and orbitals). Chemical bonds occur when an electron of one atom can simultaneously take up a vacant position of another atom.
For example, a neutral atom of carbon has six electrons which balance the charge of its six protons; two in a complete inner shell and four in its outer shell. The outer shell of carbon can hold up to eight electrons. A neutral atom of oxygen has eight protons and eight electrons; two in the inner shell and six in the outer shell (which can also hold up to eight electrons). So, an atom of carbon can bond with two atoms of oxygen as follows: two of the carbon atom's outer electrons are shared with one oxygen atom, allowing the oxygen atom to "complete" its outer shell, and the other two of the carbon atom's outer electrons are similarly shared with another oxygen atom.
When a carbon atom and two oxygen atoms are bonded like this, the electrons are in a lower energy state than if each of the atoms existed in an independent, neutral state. Different configurations, such as two oxygen atoms bonded to each other, or a cluster of carbon atoms bonded to each other also each have different energy states (of their outer electrons). Plants have complex arrangements of special molecules which allow them to use energy received from sunlight to break the bonds of carbon dioxide and water molecules to build sugars and release molecular oxygen. Breaking the carbon dioxide and water bonds is all about elevating electrons in those atoms to higher energy states. This is how/where the energy is stored. When carbon and oxygen combine to form carbon dioxide, the electrons drop to lower energy states, which means energy is released.
As plants decay into substances like coal, bonds are reformed and some energy is released - that's what drives the decay process, but the bonds between carbon atoms in coal and in molecular oxygen are still at a higher energy state than in carbon dioxide.
There is one other notion here about activation potential. The resulting oxygen and sugar molecules are stable (they don't immediately react and go back to carbon dioxide and water) because it takes some amount of energy to initiate the transition. It's like a hill with a small dip at the top, you can roll a ball up the hill over a lip and into the dip and it won't roll back down again unless you first lift it back over the lip.