Whatever the properties Arnica might have as an herb, after the extreme dilution involved in ^the preparation of homeopathic "medecine", the little sugar pills generally do not carry even a single molecule of the actual supposedly active ingredient.
In essence the pills are pure sugar.
Their "effectiveness" has never truly been scientifically tested, much less proven.
The theory is that the trace presence of the active ingredient is somehow "remembered" by the water, and this memory is transferred to the sugar. Personally I think that if it works at all, its all placebo effect.
Homeopathy's side of the story.
What are the medicines?
Homeopathic medicines are drug products made by homeopathic pharmacies in accordance with the processes described in the Homeopathic Pharmacopoeia of the United States the official manufacturing manual recognized by the FDA. The substances may be made from plants such as aconite, dandelion, plantain; from minerals such as iron phosphate, arsenic oxide, sodium chloride; from animals such as the venom of a number of poisonous snakes, or the ink of the cuttlefish; or even from chemical drugs such as penicillin or streptomycin. These substances are diluted carefully until little of the original remains.
A plant substance, for example, is mixed in alcohol to obtain a tincture. One drop of the tincture is mixed with 99 drops of alcohol (to achieve a ratio of 1:100) and the mixture is strongly shaken. This shaking process is known as succussion. The final bottle is labeled as "1C." One drop of this 1C is then mixed with 100 drops of alcohol and the process is repeated to make a 2C. By the time the 3C is reached, the dilution is 1 part in 1 million! Small globules made from sugar are then saturated with the liquid dilution. These globules constitute the homeopathic medicine.
Although such infinitesimal quantities are considered by some to be no more than placebos, the clinical experience (produced by homeopathic octors) shows that the infinitesimal dose is effective: it works upon unconscious people and infants, and it even works on animals.