How does the digestive system....

take the energy out of the food?

This is not a one-answer question.

It starts with glycolysis, citric acid cycles, electron tansport, oxidative phosphoryation and ATP.

Lets start with our digestive track which breaks down starch and other sugar containing compound to molecules which can be absorbed into the blood stream. These high energy molecules, which are broken down (in our cells) by chemical reactions, release energy and this is used to create a molecule ATP. This molecule donates it energy whenever you need energy (example muscle contractions).
But again again to answer the real how, you needs basic organic chemistry and biochemistry, so look into the pathways suggested above and possible rephrase your question more exact.

So to answer it not the digestive system which which takes the energy it only breaks the large molecule to smaller ones so they enter the bloodstream. The energy production is done by every cell itself.
Think of it as bacteria (which is not correct but to give an idea) the can get energy out of sugar, just like our cells can. But our for example liver cells do not have access to sugar. So we have the digestive track breaking down the macromolecules in our food and the blood to transport the sugar to the liver cells and the liver cell extract the energy from the sugar and create ATP. And this ATP is used for the cell functions.