The documented history is unusually clear for an esoteric discipline, because the sources survive.
It begins in Mesopotamia with celestial omen-reading, a state practice concerned with the king and the country rather than with individuals, recorded on tablets across the second and first millennia BCE. Individual birth charts appear only in the last few centuries BCE. What turned omens into a system was the encounter with Greek geometry and Egyptian practice in Hellenistic Alexandria from roughly the second century BCE, which produced the apparatus described above, signs, houses, aspects, rulerships, essentially complete and essentially as still used.
Ptolemy's Tetrabiblos, in the second century CE, is the surviving attempt to give the system a physical rationale, and it is a naturalistic one: the planets act by heating, cooling, moistening and drying. That is a causal theory, and it was abandoned as physics long before astrologers abandoned the practice, which left the discipline arguing about what its own mechanism is supposed to be. Arabic astronomers preserved and extended the material through the medieval period; it returned to Europe by translation, was standard learned knowledge through the Renaissance, declined among the educated from the seventeenth century, and was revived in a psychological register in the twentieth.
Two internal disputes remain live. The first is tropical versus sidereal: because of the precession of the equinoxes the tropical signs have drifted roughly a full sign away from the constellations that named them, so a person with the Sun in tropical Aries generally has it against the stars of Pisces. Western astrologers hold that the signs are seasonal divisions and precession is therefore irrelevant; Indian astrologers hold the opposite. The second is the outer planets, Uranus, Neptune and Pluto, unknown to the tradition before 1781, whose meanings were assigned in the modern period and whose inclusion traditionalists dispute.