I suspect that if there were a quick and easy answer to this question they wouldn’t be the powerful tools to nature they are. Just for the record, I am talking this time about how they work for us rather than how they work for the plants they come from!
So first we need to acknowledge that there isn’t a single simple answer to this question. Perhaps the most powerful mode of action is through associations and memories associated with the oils. When we have pleasant memories of oils, associated with either foods we like or indeed individuals or situations we have enjoyed, that has a powerful effect through our brains.
Clearly there is more than this however. Some smells seem to be enjoyed certainly by the vast majority of my clients if not the population. Some oils are found by most to be invigorating while some others are found more relaxing. Again just as with memories, the cells that detect scents send powerful messages to our brains. Many of the citrus oils for instance are described as uplifting with their fresh clean aroma. The citrus oils along with those derived from culinary herbs will often get the gastric juices flowing and stimulate peristalsis. (Certainly a large part of why they are suggested for constipation and other gastric disorders.)
Other oils such as Rose and Jasmine work on other parts of the brain, and they can be experienced either as sensual or nurturing depending on the situation.
To illustrate how situations can change the same physiological experience, if we look at excitement and fear, the physiological changes in the body are the same for both and yet one can be a fun experience and the other awful! Of course some may be a combination of the two such as the fear/excitement felt on a roller coaster ride. This last (not one of my favourite experiences) perhaps shows how close these two states are!
Some of the more stimulating oils have a direct physiological effect increasing localised blood flow Ginger or Peppermint. Others have an anti-inflammatory action e.g. Roman and German Chamomile.
There are many thousands of chemicals between the different oils I use (I usually have between fifty and sixty at any one time.) and these get absorbed through the skin as well as being absorbed through mucous membranes when we inhale the scent. Some of the effects of these such as the anti-inflammatory oils or those that stimulate increased blood flow or the ones like eucalyptus that can help clear blocked nasal passages are well understood. However some of the chemicals involved have never been studied in great depth. They have all been tested for safety and it is possible to look up which ones need to be avoided during pregnancy, epilepsy etc. but many of the plants and their oils have been used for centuries with literature on their therapeutic uses going back centuries or even thousands of years. Some of these uses have not been researched to a standard that would be recognised by Western medicine but may still be of use.
Another way of looking at the actions of the oils is through the eyes of Traditions Chinese Medicine and other energetic systems such as the Ayurveda.
I do not see this as replacing a Western view of things but rather complementing it. Many of these descriptions of the effects of essential oils and herbs in my view probably originated as a way to explain their therapeutic actions before our current understanding of anatomy and physiology was anything like complete!