I think I can add something to other people's explanations. Some people have mentioned that higher pressure means larger stored energy (potential energy). This is correct. And also that this pressure is responsible to accelerate the particles to the higher velocity at a narrowing. This is also true. However, it is not correct to say that the lowering in pressure comes as a result of the conversion from potential energy to kinetic energy. That is, it is not that the particles have less kinetic energy (which would mean lower temperature).
What actually happens is that the particles are moved on average further apart in the higher velocity zone (see figure below)
The figure represents a 1-d model of a train of vibrating molecules travelling from, say, left to right at a constant velocity (top image) until they encounter a zone of higher velocity (lower image). The circles represent a sample of the particles' positions, such that more likely positions have more circles clustered near them.
Just as in the flow of molecules in a pipe, one must imagine the molecules vibrating back and forth. This vibrating motion is superposed over the average velocity left-right. The temperature of the fluid gives the average kinetic energy of the molecules due to the random motion (subtracting the component due to the average velocity). When the particles pass through a zone of higher velocity (such as a narrowing in a pipe), they become more separated, making the frequency of collisions smaller. This is mainly what causes the lowering of pressure, since a smaller number of impacts per unit time and area leads to a lower total force per unit area.
It is true, as others have noted, that the particles use part of their kinetic energy to accelerate, lowering their temperature slightly. However, this is normally a very small fraction of their vibrating velocity, so that the change in temperature is small. The change in distance between the particles is the real cause of the drop in pressure (potential energy turns into kinetic energy).
I hope this explanation helps you and others understand this commonly asked question. I would also like to point out that this question has been asked in this site, with variations, in other occasions:
How can we intuitively understand the idea that when the velocity of fluid increases, the pressure of fluid decreases?
Why is pressure greater in an open part of a tube than in a constricted one?
Also related is
Why decrease in velocity will increase pressure?
Complementary explanations are given in
Microscopic source of pressure in an incompressible fluid