My understanding of it is that when light passes through a medium, it does not do so in a straight line. So, while the actual speed of light remains constant, it takes longer to get from one end to the other.
To be precise, light can travel through a medium in a straight line, but it bends when it passes from one material to another.
The light bounces off the inner walls of the fiber in a zig-zag fashion as it propagates, so it cannot possibly travel at the speed of light. It depends on the index of refraction as mentioned.
