This is a common myth that has a small teensy bit of truth to it.
When you shave, you are not removing the hair, you are shortening it to skin level. Compared to growing a new hair, your cut hair has a head start:
1. It is an active follicle
2. It already has some hair grown, just you cannot see it below the surface of the skin.
There is also the myth that your hair grows back thicker after being shaved. There is also a small amount of truth to this as well. When a hair grows in, it is not a cylinder. It is in fact an elongated cone. The tip of a new hair comes to a point and as it grows it gets thicker until it reaches its full thickness. When you cut the hair off at the skin, you have cut it off at its full thickness. Now when it starts to grow in, it will be full thickness to begin with. It will not be any thicker than before, but it will already be at full thickness. This is also why hair stubble is very stiff when compared to new or longer hair. The short hair being already at maximum thickness is a lot stiffer. Fresh hairs are still very thin and bend easily.