First consider a pot with a loose lid over it. As the temperature rises, the vapor pressure of the steam rises as well (even before boiling) and the steam will push its way out under the lid. You'll get cooking temperatures over 100C, but not by much, and a lot of the energy you expend on cooking just goes out into the air instead of into your food. And to add insult to injury, your lid will rattle around on top of the pot.
Now consider the pressure cooker. The lid is well sealed and tightly locked down. The vapor pressure rises with temperature as before, but now it has nowhere to go; the pressure can't relieve itself by pushing the lid up and escaping. The energy you put into heating the food and water stays in the pot, and consequently you can achieve higher temperatures inside the pot. In fact, the "pressure" part of a pressure cooker is only a side effect of trapping the hot gasses inside the pot; the real benefit is the higher cooking temperature.