Vibrational Essences work on the principle that the energy of a plant or crystal (or whatever the essence is made from) can rebalance your emotions when it makes contact with your body (the process can take a few weeks). It does this by creating a resonance with your energy body (your aura). It's a gentle, safe, and effective tool of healing, and is particularly good for emotions such as grief, anger, sorrow, jealousy and envy, loneliness, bitterness, regret, and so on.
The principle of shamanic healing (and all types of spiritual healing) is that the physical manifestation of a disease is the last manifestation. So before it becomes the ulcer in your stomach, or the tumor in your lung, it is a blockage - a negative energy - in your outer body. These blockages then start to manifest themselves as negative and unbalanced emotions (and then finally, they will manifest physically as an illness). Vibrational Essences work on this level, healing you from the energy and emotional bodies, inwards.
Rescue Remedy is the most well known flower essence in this country, created by Dr Edward Bach in the 1930s as a way to heal acute shock and trauma. When he studied the flowers, he associated their external, visible characteristics with the human emotions the flowers could help to rebalance, thus bringing a person back to the state of the positive (e.g. Impatiens grows very quickly and very tall, therefore it is able to rebalance impatience, impulsiveness, and the need to be first all the time). He was also able to see how emotional imbalances led to physical illnesses (anger can lead to liver problems; fear can lead to kidney problems; and so on). Once the emotions have been rebalanced and brought back to the positive, the physical body will then heal itself. This was the basis in which vibrational essences in the western world was then built.
Shamanic cultures around the world have been using essences in similar ways for all time, e.g. petals soaked in water as an offering, as a drink, to cleanse and purify the system, strengthen the body, and to connect us to our higher selves. That petal-soaked water would hold the energy of the flower, just as flower essences do.