The fundamental reason to root prune is to maintain and maximize the efficiency of the root system of the tree while minimizing the problems that growing in containers presents.
As the roots grow, they change from feeder roots (thin roots) to anchoring roots (thick, woody roots). As they become thicker, their efficiency in taking up moisture/nutrients begins to wane. Anchor roots are great for trees grown in the ground. The last thing any (most) of us want is to see our large fig trees blowing down the road during a wind storm because the roots did not hold it in place. In containers, however, the anchoring roots become less necessary and even counter productive. Given that most growing mixes occupy about 50% of the volume of the container, and the water we put in occupies part of the remainder, there is a great benefit to removing less efficient roots thereby allowing more room for the more efficient (feeder) roots to occupy. Root-pruning containers seem to do a decent job of accomplishing a dense, fibrous root-mass.
Root pruning can be accomplished in many different ways. There are, however, benefits to periodic bare-rooting of container grown trees. When a tree is freshly potted, it is occupied something like 50% growing mix / 25% air and water / 25% roots. Over time, the roots of the tree grow and the components of the growing mix decay and the ratio changes (I'm making up numbers for illustrative purposes only) and it may look something like 30% / 15% / 55% and the tree is still grows in a healthy manner. Given enough time, though, the ratio becomes more critical... getting to a point where the roots and remaining growing mix occupy 100% of the container. The growing mix has decomposed and compressed so much when the tree is watered, the water runs down the side of the container and out of the drain holes. Even at this stage, the roots do not stop growing. As the root mass compresses, the roots will be girdled. Simply moving the tree into a larger container does not alleviate the compression. Period bare-rooting eliminates the compression and keeps the growing mix fresh (particle size is important). It also allows the grower to identify and correct problems.
In 2006 I bought a ~6' tall Black Mission fig tree in a 3 gallon pot from a nursery in NJ (it is one of the trees in the car behind me in my avatar picture). The tree was long overdue for root work and barely held water. Before I did root work on it, I took a chopstick and drove it into the top of root ball as hard as I could. The chopstick did not go in more than 1/4". It took me more than 3 hours to bare root the tree and it was evident when I got to where the side of the one gallon container and the rooter container were and each one was more difficult to work through. This is not the only tree I've received with this type of compression. This past year, I bought some trees from a nursery. One in particular I noticed small particles of growing mix washing out through the drain holes when I watered it. When I pulled the tree from the container, the bottom 60%+ was nothing but roots.
Here is my method for bare-rooting and root-pruning a tree which I have not done root work on before: I water the tree about 24 hours before I start. Once I remove the tree from the container, I use chopsticks to start removing the growing mix and comb out the roots from the bottom up. Once I get through the bottom inch or so, I start raking the outside of the root ball. Once I can get the roots pulled away, I work (along with a little shaking) the growing mix out of the root ball. In general, I work from the bottom up. Once all of the growing mix is off, I cut the roots to 1/3 to 1/5 the radius of the container the tree is going into. Then I remove a few (it's a judgement call but probably about 20%) of the ticker roots from the base. Then into the container with new growing mix... water in and stake. Here is a before and after on root pruning of an LSU Purple:
