How did democracy first develop in Athens?

Athens was originally ruled by kings, who controlled the surrounding region of Attica. By the eighth century BCE, rule by kings had developed into rule by archons (literally "rulers"). The archons were members of the aristocracy; they each served a particular function in ruling the city, and they arrived at decisions by committee.
In the 630s BCE, an aristocrat named Cylon attempted to overthrow the archons and make himself sole ruler of Athens. The other aristocratic families murdered him, and there was social unrest in the city for a while.
About a decade later, the Athenian citizens asked another aristocrat, Draco, to be their ruler and lawgiver. Draco reformed the existing laws of Athens and codified the new ones so that they could be referred to and enforced by a court. He replaced the small council of archons with the Council of Four Hundred, a group of four hundred citizens who worked with the archons to defend the city and enforce the laws.
Draco was a harsh ruler and was eventually exiled from Athens. Another period of social unrest followed, and the Athenians appointed another ruler, Solon, to revise the laws. Solon established the Athenian Constitution, which included the right of any Athenian citizen to a trial by jury and the selection of officials by lot (at random) to avoid the concentration of power in the hands of any particular group. These were prerequisites for the development of Athenian democracy.
Following Solon's death, there was yet another period of social unrest, at the end of which an aristocrat named Cleisthenes emerged as the ruler of Athens. Cleisthenes enacted yet more reforms to Athenian law and political life, the aim of which was to break the power of the aristocratic families and bring all citizens to the same "level" so that no one man had more power than any other. To prevent any person from acquiring too much power, Cleisthenes invented the practice of ostracism, whereby the citizens could vote to exile a dangerous (or unpopular, or overly powerful) individual from the city for a period of ten years.
The various reforms over the years, and particularly those of Cleisthenes, led to the dissolution of the old rule by archons and the establishment of rule by the citizenry, or assembly (ekklesia), which consisted of all free Athenian men over the age of eighteen. Members of the assembly could vote on any topic that affected the citizenry, including military spending, diplomatic matters (treaties, envoys to other nations), allocation of resources (food and money), and the creation of new laws.
You may find the following resources useful:
Aristotle: The Constitution of the Athenians
The Development of Athenian Democracy
Democracy in the Politics of Aristotle