How is the digestive system linked to the circulatory system?

The digestive and respiratory systems are closely linked, and as with all other body systems one cannot function properly without the other.
First, think of the functions of each system separately to help you understand how each system relies on the other.
The digestive system breaks down food to provide energy to the other organs and organ systems in order for those things to do their jobs.  At the cellular level, the digestive system is providing glucose to the cells.  The mitochondria need this glucose as one part of the equation in cellular respiration.  In simple terms, this formula is
Glucose + Oxygen  -----> Water  + Carbon Dioxide + Energy (ATP)
So the digestive system provides one of the things you need in order for your cells to produce energy.
The oxygen is brought into your body by the respiratory system.  Your circulatory system is the delivery system for your body.  It delivers the oxygen from your respiratory system to your cells as well as the glucose from the digestive system.  
Without the circulatory system, your cells could not produce energy because they would not receive the glucose and oxygen they need to function.  Without the glucose provided by the digestive system, your cells could not produce ATP and the circulatory system would not have the energy it needs to do its job. So, the circulatory and digestive systems are closely linked and important to how your body functions.

The circulatory system is actually linked to all of the systems because it is responsible for transporting oxygen, nutrients, hormones, and cellular waste out through the entire body using blood vessels. The digestive system is responsible for converting food into nutrients and other vital components so that the circulatory system can transport them to their proper location in the body. The digestive system begins with the mouth (ingestion) and ends with the large intestine (excretion). In between those points the other organs--pharynx, esophagus, stomach, small & large intestines, pancreas, liver and gallbladder work together to filter food, decide what to keep and what to discard, and transport. It is in the large intestines specifically where the food is broken down and nutrients are absorbed into the blood (circulatory system).

The digestive system is made up of various organs that are all supplied blood by the circulatory system; however, I don't think that is the intended connection that the question implies.  Food enters the digestive system through the mouth, and a combination of mechanical and chemical digestion begins assaulting the food there.  After swallowing, the food travels down the esophagus to the stomach.  Chemical digestion occurs there over the next few hours, and then the food (known as chyme) moves from the stomach into the small intestine.  Chemical digestion continues to occur in the first section of the small intestine (duodenum).  Once the food is fully broken down, it needs to be absorbed into the blood stream in order for those nutrients to be transported to cells all around the body.  This is where the digestive system and circulatory system are linked.  The middle section of the small intestine (jejunum) is covered in tiny folds called villi.  These folds increase the surface area and absorption space of the small intestine.  Each villi contains specialized cells that transport different nutrients into the blood.  Once in the blood, the circulatory system delivers the nutrients to the cells. 
https://livehealthy.chron.com/importance-villi-small-intestine-digestion-nutrients-3923.html