Reins are built 'on' a steed to allow the option to improve its speed. You must mount the steed to start the reins project.
Reins only improve your speed when you use them at the project tick immediately prior to the travel tick. If you are not using them when projects are processed, the boost will not be applied when travel is processed. They show up on the objects page like an engine in a vehicle with a button to "hold" them. Once you "hold" them, you will join a project called "working at reins."
There is also an option to disassemble them.