But I wonder why the biblical people did it. There are other references to oil in the Bible -- oil lamps, for example. So oil had some kind of cultural significance. But someone had to be the first to come up with the idea of connecting oil with special ceremonies. Maybe it was a substitute for blood. Smearing blood on things goes WAY back in human history. Blood of your prey, blood of your enemies, etc.
Oil lamps and the like have spiritual significance in other religious traditions as well, particularly the Vedic traditions and the faiths derived from them (including Hinduism, select Buddhist traditions, Jainism, Sikhism, and the Hare Krishna faith). No connection with blood in these Eastern traditions. Oil is also used in Ayurveda, an ancient Indian system of medicine, for a number of health-related uses; oil massages and drops of pure oil in the nose (nasya therapy), for example, are used on a daily basis or as treatment for certain illnesses and disorders.
Oil lamps are particularly known for use during the celebration of Divali, a festival jointly celebrated by Hindus, Jains, Sikhs, and Hare Krishna devotees, known as the Festival of Light. The details of the reason for celebration differ slightly between these faith traditions. In all traditions, it is a celebration of victory over evil, of knowledge over ignorance, or of light over darkness.
Oil lamps have also been used in Pagan traditions, as well as the Vedic tradition. So they (and more generally, oil) have a long history of use in the a religious context even long before, or alongside, the Biblical use of it.
The very definition of anoint is to smear or rub with oil, typically as part of a religious ceremony--like in the Bible. But if you rub butter on the crust of your bread you are anointing it. I always anoint my turkeys while roasting.
They always said to use "the purest olive oil." That would be something expensive, virgin, and Tuscan. That's a sad waste of Tuscan olive oil. Best to use the cheap 2nd hot-pressed Spanish stuff for a dollar a bottle. Why you can't use simple vegetable or rapeseed oil is beyond me.
Did anyone have a little consecrated oil dispenser on your keychain for roadside emergencies? I had one that fit into a fountain pen I used. It really impressed people when I used it. I saved so many lives by annointing people with oil, saving their limbs and calling them back from the dead. I hate to brag, but there it is.