The graph above illustrates the state of the supply and demand for oil.
In blue, is demand for oil. This curve is almost vertical. As demand increases, this curve moves to the right. That causes a rapid increase the price.
In red, is the supply for oil. This curve becomes perfectly vertical. As supply decreases, this curve moves to the left, and the cost goes up rapidly.
With peak oil, both curves are moving at the same time, towards one another. That causes a very rapid increase in prices.
At the moment, the supply curve (red) is static, but it will eventually begin moving slowly to the left. The demand curve(blue) is currently moving to the right. This is causing the increase in the price of oil.