Bernoulli's principle states that for a fluid (gas or liquid) of very low viscosity and a poor conductor of heat, an increase in the speed of the fluid occurs at exactly the same time as a decrease in pressure.

To try and explain why that is look at the first diagram. If you were in the little blue submarine the water would have to go faster through the narrow pipe so the submarine would accelerate. To make it accelerate a force is needed and that force van only come from the water pressure so the pressure behind the submarine would have to be higher than it is in front.

Now imagine you have reached the part of the tube where it becomes wider. The water and the submarine have to slow down so the pressure in front, that is in the wide tube) must be larger that the pressure behind in the narrow tube (diagram 2). That is the basic statement of the Bernoulli Effect, when the fluid (water in this case) moves faster, the pressure is less.

Watch the video for a more thorough explanation.

Diagram 3