How is chemical energy stored in carbon
One of the most important issues in understanding bond energy in chemistry is the question: energy relative to what?

The formation of bonds from (mostly hypothetical) atomic elements releases energy (or they wouldn't be bonded.) Solid carbon (graphite or diamond) has less energy than a cloud of carbon atoms so it could be said graphite has less energy relative to a cloud of atomic carbon atoms. We would describe this energy as "bond energy". But this is the energy released by the hypothetical process of forming lists of C-C bonds from isolated carbon atoms. 

But there is another subtle complication if we want to understand reactions. sometimes, when chemists write a reaction scheme they don't specify all the detail. So C +O2->CO2 doesn't actually mean atomic carbon atoms plus oxygen molecules give carbon dioxide. Most chemists would assume that the carbon starts in the state it is normally found as in laboratories: graphite (so not isolated atoms but a molecular solid containing lost of C-C bonds). This terminology seems to have confused some people. But chemists don't like to waste space in formulae redundantly specifying things most of their colleagues will already know. The default state of an ingredient in a reaction is assumed unless specified otherwise and this usually means the form the compound or element takes in a bottle on the lab shelf.

So how can we say that carbon stores energy? Well we have to ask the question, relative to what? In the case of the question the answer is relative to carbon dioxide gas. The point here is that the double bonds in CO2 are stronger than the combined bonds in oxygen gas and graphite (so they are lower energy) This means that the reaction between carbon and oxygen releases energy (it takes less energy to break all the O-O and C-C bonds than is released when new C=O bonds are formed. So elemental carbon "stores" energy relative to carbon dioxide. 

The claim that carbon "stores" solar energy absorbed by plants is a little simplistic (mainly because plants don't store carbon but usually store more complicated molecules like sugars or poly-sugars like lignin or cellulose). Dead plants are converted to more carbon like compounds (oil is mostly hydrocarbons, but coal is mostly carbon) under some geological conditions. What plants actually do at the start of this process is to convert carbon dioxide into sugars using energy input from sunlight. They do this to store energy and to grow by converting the sugars into the structures of their leaves, stems and trunks. 

We can reverse that process by burning the plants or their degradation produce like oil and coal in air to reproduce the carbon dioxide and release energy. In this highly simplified sense, carbon is a store of "solar" energy. 

But the key to understanding what is meant is to understand the chemical reaction involved in the creation or burning of the carbon. Then you can understand what energy is stored and how it is stored.