During strenuous muscular activity, lots of ATP is demanded by the muscles in order to contract maximally. The most efficient way to generate ATP is aerobically (using oxygen) which provides 36 to 38 molecules of ATP per molecule of glucose. Thus, the muscle cell would 'prefer' only to metabolise glucose in this manner. However, at peak intensity, more ATP is required than can be generated with the provided oxygen, despite the body's best efforts at getting as much oxygen there as possible. Thus, the muscle must grudgingly metabolise glucose anaerobically (without oxygen) to make up the balance of the demanded ATPs.
The byproduct of anaerobic metabolism is lactate. In itself, this causes a lactic acidosis (indirectly), which can cause trouble for the muscle as it leads to pain, cramps and inefficient muscular contractions. More of a problem, though, is the fact that the muscle is in danger of running out of glucose to metabolise, since it is chewing it up rather inefficiently anaerobically (producing only 2 ATPs per glucose). The Cori cycle is an attempt to remedy both of these problems.
In the Cori cycle, the lactate produced by the muscle is transported to the liver. At the liver, it is pushed back up the gluconeogenic pathway, producing first pyruvate and then... glucose. This glucose is duly released by the liver into the bloodstream, where it may be used by the muscle again. If the muscle is still busy frantically contracting, the glucose is used to produce ATP, and the whole business can begin again (hence the 'cycle' part). However, should the muscle have finished its business, the glucose can be used to restock the muscle's glycogen stores.