The expression free rein originated in horseback riding.
The rider holds reins, leather straps connected to a metal bit in the horse's mouth, and controls the horse by pulling on the straps in different ways. When you give a horse free rein, you hold the reins loosely so that the animal may wander where it wants. In contrast, if you keep a tight rein on something, you control its every move.
Free rein is an effective way to say "freedom" in a sense where that freedom has been granted or allowed by a person or institution; you'll often see this phrase after some form of the verb give, as in the first example below:
Pollution standards were relaxed, and development projects were given free rein to operate in forests and other natural areas, contributing to deforestation. (Slate)
He now had free rein to explore the house. (The Marvels)
It's easy to understand why people confuse the correct expression, free rein, with free reign. Rein and reign are homophones, and both words deal with forms of control. Reign means "to rule," like a king or queen does. As a noun, it can refer to the period when a monarch rules or royal authority more generally. By definition, a monarch already has a free reign because he or she is the ruler, the supreme authority in his or her kingdom. So free reign is redundant.
Free reign is a common mistake, however, and even editors who should know better sometimes don't catch it:
Instead, the teachers are given free reign to create their own curriculum based on the things they want to learn more about. (Salon)
He's pretty much been given free reign to let his imagination run loose when he's hired. (Los Angeles Times)
If free reign is wrong, then so is reign in, another commonly misused phrase. To steer clear of any confusion, remember that the correct expressions, free rein and rein in, have their roots in horseback riding.