How does a plant cutting develop roots?

Plants grow only from regions at the tips of the roots and shoots called meristems.

Within the meristem areas there are stem cells ("blank" unspecialised cells). Unlike animal stem cells, plant stem cells are totipotent - meaning that they can differentiate into any type of cell. Therefore when the cutting is taken from the end of the shoot, the stem cells can differentiate into root cells or shoot cells depending on their conditions.

Because the meristems (therefore the unspecialised stem cells) are only located at the tips of shoots, you cant grow a cutting from the middle of a branch.