Teasing or tickling the roots of a plant refers to loosening pot-bound roots. Many plants we bring home from the nursery have been growing in the same pot for months. They look fine on top, but when you slide the plant out of its container, the roots are a circling, tangled next. Loosening the roots before planting in the ground will allow you to spread the roots out in all directions, so they will branch out in the soil and from a good foundation for the plant. This is a good planting technique for all plants, but it is crucial for plants that are pot bound.
Very often when you purchase a plant in some type of pot or container, it has been growing in there for a good amount of time, sometimes over a year. What started as a small seedling has grown, not just on top, but also under the soil line. The roots of a plant can grow considerably in size, very often more so than the foliage of the plant. Once they completely fill the container they are growing in, they have nowhere else to go but around in circles, inside the pot. When that happens, the roots start to circle and tangle around themselves, looking for somewhere to escape.
When a plant is pot bound, you will probably see some roots poking out the drainage holes and maybe even through the top edges. It may seem like a bonus to purchase a plant with such a robust root ball, but in actuality, having its roots wrapped in a knot is very stressful for a plant; so much so, it may never recover, leaving it stunted or killing it outright.
If you place a pot-bound plant into the ground or into another pot without first loosening the balled up roots, they will continue to grow in a circle, rather than reaching out into the soil, developing and anchoring the plant. So although it may seem like a harsh thing to do to your plants, you are actually helping them by roughing up the root ball.
In this case, the kind of teasing is very similar to what is done to hair; you comb your fingers through the roots to loosen them and increase their volume. Do this just before you are ready to plant. In most cases, you can loosen the roots with your fingers. Try to be as gentle as you can, but it's okay if a few roots get broken in the process. Better to have a couple of small, damaged roots than a bunch of intact roots that are strangling each other.
If the roots are so tight that you can't get your fingers between them, try soaking the entire root ball in water for a few hours, or overnight. Very often they will begin to float apart, making it easier for you to work the remainder of roots apart with your hands. You don't have to loosen every root, but try to ease apart as many as you can.
In extreme cases of pot-bound roots, you may need to slice through the root ball with a sharp knife or pruners. Do this in several spots around the root ball, to encourage root growth in all directions. It may seem harsh, but the plant will send out new feeder roots and should soon recover.
Annual plants, such as bedding plants and vegetable seedlings, are grown in very small containers and often come with a knotted mess of young roots that are easy enough to tease apart. However, many gardeners don't bother with annuals, since they are only going to grow for one season and don't need to develop a strong root system. It is perennial plants, and especially trees and shrubs, that receive the most attention and not be allowed to remain root bound.
Whichever method you wind up using, be sure to give the plant plenty of water, both when you first put it in the ground and for several weeks afterward. Damaged or traumatized roots require time and a bit of TLC, to heal.