Priests in Ancient Egypt could predict eclipses and the floods of the Nile by observing the stars and the Moon and recording their previous positions when the events of interest happened. The rest was calculation, nothing magical. Ordinary people saw the priests looking at the stars and predicting events in the future, and thought that the stars magically told priests things and that the prediction ability extended to all future events (births, deaths, outcomes of battles). The priests encouraged this belief, because it gave them more power. This is one way astrology could have developed – by distorting and exaggerating the science of astronomy. Another way is via navigators telling the latitude of a ship using the stars or the sun. People would have thought that if heavenly bodies could tell a navigator his location on the open sea, then why not other secrets?
Engineers in Ancient Rome calculated the strength of bridges and aqueducts, and estimated the amount of material needed for these works. Ordinary people saw the engineers playing with numbers and predicting the amount of stones needed for a house or a fort. Numbers “magically” told engineers about the future, and ordinary people thought this prediction ability extended to all future events. Thus the belief in numerology could have been born.
When certain plants were discovered to have medicinal properties against certain diseases, then swindlers imitated doctors by claiming that other natural substances were powerful cures against whatever diseases. The charlatans and snake oil salesmen distorted and exaggerated medicine.
Doctors diagnosed diseases by physical examination before laboratory tests were invented. Thus a doctor could look at parts of a person’s body, tell what diseases the person had, and predict the symptoms that the person would experience in the future. Exaggerating this, palm readers claimed to predict a person’s future life course by looking at the skin of their palm.
In the 20th century, some medicines were discovered to be equally effective at somewhat lower doses than previously thought. Then homeopathy exaggerated this by claiming that medicines are effective when diluted so much that on average not a single molecule of the drug remains in the water given to the patient.
In all these cases, superstition only adds bias and noise to scientific results. Science does not know everything, but it is a sufficient statistic (https://en.wikipedia.org/wiki/Sufficient_statistic) for superstitious beliefs, in the sense that any true information contained in superstition is also contained in science. Nothing additional can be learned from superstition once the scientific results are known.