According to an Ohio State University study, they really don’t.
Aromatherapy claims can be found in everything from candles, lotions, bath salts and more when certain essential oils are added to the product. The claim depends on the essential oil that is used – lavender to help you sleep, basil for headaches and lemon as an anti-depressant. OSU conducted a study using lemon and lavender oils and put the subjects through several scenarios from pain response (dunking feet in cold water) to mood response (completing psychological tests). What they found was that lemon oil had a positive effect on subjects’ moods, but lavender didn’t have any perceived effect. Neither essential oil significantly affected pain responses, heart rate or blood pressure.
I doubt these finding will curb much, if any, spending on aromatherapy products, but the findings are at least interesting. I personally think if you feel it works then it’s worth it. Whether there are real results of that essential oil or you psychologically feel the effects, then really it does “work.” It just may not have the same effect for everyone.