How do plants convert light from the sun into energy?

The short answer is photosynthesis! But what does that really mean?

In a nutshell, plant cells have organelles (the "organs" or a cell) called chloroplasts that have green pigments called chlorophyll located inside of them. The chlorophyll are arranged in a way that allows the energy received from sunlight to excite their electrons and pass an electron from pigment to pigment until it reaches the reaction center (a special chlorophyll A). The above mentioned is referred to as the 'light dependant stage.' The very first electron is donated by a water molecule, splitting it into hydrogen ions and oxygen molecules (which plants release for us to breath). This is why plants need water!

This electron is then passed along an electron transport chain consisting many pignments embedded in the thylakoid membrane (little stacks within a chloroplast). This chain uses electrons to form ATP (potential energy stored in a chemical compound) and NADPH (an electron carrier). They will come back into play in a short bit...

The 'dark' or 'light independent stage' starts when the carbon dioxide absorbed by plants from the atmosphere enters a series of reactions collectively called the Calvin Cycle. Through a handful of reactions, carbon dioxide is transformed into various compounds and then back again, in a cycle. The ATP and NADPH created in the transport chain earlier provide the electrons needed to change up the compounds. Half way during the cycle, glucose is released. This is the 'sugar' that plants produced. For example, the sweet taste of fruit. This is is essentially the food 'energy' that humans need to survive.

To go beyond your question a bit, the energy that humans get from eating plants comes from the opposite process of photosynthesis; cellular respiration. Here, through a number of different chemical reactions, the glucose molecules are broken back down and give ATP (remember, this is the potential energy molecule). ATP (or adenosine triphosphate) releases lots of energy when it is broken down into ADP + P.

There you have it - how plants convert light from the sun into energy!