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Cells use chemical energy. Some organic chemicals are rich stores of chemical energy. Why is this?
Answer 1:

All molecules are made up of atoms which are joined together by bonds. The bonds which join atoms contain energy. Because different atoms have different properties, the bonds between different atoms will vary in their strength and amount of energy.
The cells in your body function by manipulating and rearranging molecules to form new bonds or break old ones, in a process called metabolism. By controlling the formation and breakage of particular chemical bonds your cell can control the energy present in the bonds. Basically, everything that happens in your body is the result of these controlled chemical reactions that are occurring in your cells.
Some chemical reactions release energy when they occur. Others require energy. Your cell is able to use chemical reactions that release energy in order to achieve desired reactions that consume energy. It accomplishes this through an intermediary molecule called ATP. ATP is a molecule that contains three phosphates, and the bond between two of the phosphates contains a lot of energy. Your cells use the energy released from an energy-releasing chemical reaction to produce ATP (by joining the molecule ADP with another phosphate molecule). By coupling these reactions the cell has effectively captured some energy in the bonds of the ATP. (This is just one way of producing ATP, called substrate-level phosphorylation. But there are other ways that cells can produce ATP too-- for example, through aerobic respiration or photosynthesis!). Later, when the cell needs energy to perform another chemical reaction, it can use the energy released by breaking the phosphate bond in ATP (again separating the ADP and phosphate molecules) to make the reaction go.
Some organic molecules contain a lot more energy in their bonds, due to the types of bonds they possess. This makes them rich stores of chemical energy. Keep in mind, though, that it's not just the energy in the bonds that is important to your body! The bonds must also be of the right type for your body to use in its chemical reactions!
If you want to know why, exactly, some bonds contain more energy than others, that is something that unfortunately I do not understand. I hope a chemist or a physicist can help explain that to you!

Answer 2:

It takes lots of energy to make some organic chemicals.Some of this energy is lost as heat when the organic chemicals are synthesized, and some of this energy is stored in the organic chemicals. When these organic chemicals are broken down or metabolized into smaller organic chemicals, some of their stored energy is released in forms that can be used by the cells.
A mechanical analogy is that you can roll a stone up a hill, which needs energy, and puts the stone in a position of higher energy. Then you can let the stone roll down the hill, where it can do work along the way, such as by knocking things over.
One organic chemical that has a lot of chemical energy is ATP [Adenosine TriPhosphate]. When it loses a phosphate group and becomes ADP [Adenosine DiPhosphate], it releases a lot of energy that can be used to create new chemical bonds.

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