Your circulatory system is made of three main "sections." Your heart, blood vessels, and blood. In other words, the pump, the tubes, and the liquid. I can't really explain exactly what happens to your circulatory system during athletics without involving the respiratory system, which is composed of your lungs and the tubes and muscles associated with moving air in and out of your lungs.
When you begin playing football (or any exercise for that matter), you are asking your cells to produce and use more ATP. ATP is the unit of energy that your cells use. Think of it like a battery for every cell in your body. More muscle contractions means more ATP is being used. More ATP usage means more ATP production demand.
In order to make ATP your body goes through a process known as cellular respiration. The basic formula is Glucose + Oxygen = Carbon Dioxide + Water. Up to 36 units of ATP energy can be produced in this way. What the formula tells you is that during exercise every single cell is asking for more oxygen to be delivered and more CO2 to be taken away. Oxygen is coming from the lungs and carbon dioxide needs to be taken to the lungs. The only pathway that this can happen through is the blood and blood vessels. In order to move the blood more quickly through those vessels, two things happen. Your vessels dilate and your heart rate increases. This moves blood and the gasses it carries more quickly.
So as you can see your circulatory system and respiratory system are closely linked in order to provide each cell with the materials that it is demanding in higher amounts while exercising.