With long exposure photography the sensitivity of the sensor is much lower so it requires a longer exposure time to the scene in order to absorb enough light to form a clear image. Usually with long exposure photos there is some stationary objects and then there are some (usually brighter) moving objects. It takes the entire length of the exposure to capture the dark parts of the shot like the mountains in the background but since they are not moving at all, they still come out fairly sharp. The LED lights are by far the brightest thing in the image and since they are putting out so much light, the sensor is able to capture that even though it is moving quickly across the scene.
I'm surprised there is no hint of the kayaker in the shot, but I guess they were dark enough that the camera didn't pick them up. You can even see how it works by looking at the water, the ripples in the water move pretty quickly so that causes a lot of blurring which is why most of the lake is a solid blue color. But as the kayaker went by with the light it reflected off the water which gave the camera enough light to capture more detail which you can see. The detail in the water at the left side of frame and the detail in the right side of the frame is actually from 2 different times, probably several seconds apart.
These images were created by mounting some LEDs to the kayak paddle which is how you end up with the pattern you see there which is actually the path the paddle takes as the kayaker glides across the water. You can read more about the photographer and the equipment he used from the link I posted.