The work done at the barrel is equal to the work done at the nozzle.
Word Done = Force x Distance.
The force at the barrel > the force at the nozzle, so for equal work done,
distance moved by water leaving barrel per second < distance moved by water leaving the nozzle per second.
This results in a greater speed of the water leaving the nozzle.
Alternative Answer:
Volume of water leaving the barrel per second = Volume of water leaving the nozzle per second.
The cylindrical volume = Base area x distance moved by the water.
Since base area of barrel > base area of nozzle, so for equal volumes,
distance moved by water leaving barrel per second < distance moved by water leaving nozzle per second.
This results in a greater speed of the water leaving the nozzle.