We have evidence of trepanning (drilling holes in the skull) going back to the flint tools time period. We still use this today to release pressure after a bleed in the skull.
It was a lot more brutal and had a much lower success rate back then. But the fact that we find so many skulls with evidence of trepanning means that prehistoric humans must have considered the low success rate worth the risk. What's interesting is there's no way they actually knew what trepanning could help with, since it's to do with intracranial pressure. So in the same way the medieval cure for everything was bleeding, whether or not the disease had anything to do with blood, trepanning seems to have been the proverbial hammer for which everything looks like a nail.
Head injuries would have been common, bleeding on the brain was probably easily recognisible in warriors for which it would help. How they discovered that it helps.. nice
I mean, even on a mild headache you feel like the inside of your head is swallen, even if you don't know what a brain is. Imagine a traumatic fall, fever and constant pain. After the tenth time you see that happening you grab a stone and try nad fix it.
Chimpanzees are known to put special plants on wounds to heal them better, so my guess is that other animals do it also to some degree. Cows eating special plants for their stomach, chickens eating small rocks and sand, hell even dogs and cats eating some plants to fix their stomach.
Go check out the alledged link between the snake wrapped staff that's used to represent medicine and the treatment for guinea worms. Googling puts that theory with the Ebers papyrus from 1500 BC if it's true!
Confusingly, there's actually two similar staves that get mixed up. The helix patterned one with two winged snakes I think you have in mind is called the Caduceus, but the the single wingless version I meant is the staff of Aesculapius (multiple spellings out there).
The Edwin Smith Papyrus, a papyrus made to treat traumatic brain injury or TBI, is the very first written work made to remedy a medical condition in a way that doesn't depend on sorcery, written around 2000 BC. It gives a detailed account on some but not all do's and dont's of such injuries. I cite this because it actually suggests Egypt knew better than those of us alive today.
Animals were doing it long before humans even existed. Some birds will "bathe" in an ant nest because the formic acid excreted by the ants rids them of parasites. There's even a word for it - zoopharmacognosy.
Long before recorded history, people knew what plants were helpful to treat or cure various maladies. Who knows what possessed the first human to chew on willow bark to relieve pain or reduce a fever? The earliest documentation of it was 400 BCE by Hippocrates, but it was probably common knowledge for much longer than that. The Chinese have been using various herbs to treat disease for at least 3000 years.