This myth is actually not true. 

This works well in the garden, but not on the scalp. Unlike the stem and branches of a shrub, the shaft of the hair is not alive, so ‘cutting back’ will have no effect except to give the appearance of thicker hair when it is shorter.

Once hair is visible on the surface of the scalp, all you can influence is its condition and appearance.

The science bit:

Hair grows from the dermal papilla, or hair root, which is located at the base of the hair follicle. The only factors that influence the forming hair cells within the root are genetic predisposition, diet, illness, physical trauma etc., will determine how strong or weak the hair will be.

Once hair has emerged from the follicle its ‘strength’ has been determined by its diameter, elasticity and tensile properties are set and cannot be influenced by cutting it. What does now start to influence the newly emerged hair is the environment and the things we do to it such as washing, colouring, straightening etc., and if weakness and damage is caused to the hair from these factors then cutting off any split or broken hair certainly is advantageous.