The short answer is well controlled studies do not show the benefits that many of these products claim. It’s important to know that the FDA doesn’t regulate over the counter supplements and herbs, and that they are allowed to claim health benefits as long as the seller doesn’t claim they treat disease. So you can see claims like “supports a healthy immune system.”
There’s a comprehensive article about essential oil in pediatrics here.
Pleasant scents like lavender may make you feel more relaxed. And the placebo effect is high–a third of people report improvement to many sugar pills and even sham surgery. An excellent book on complementary medicine is Snake Oil Science by R. Barker Bausell.