To better get an idea as to what challenges Dracon and Solon were responding to, and the nature of the instability they faced, it's helpful to look at the historical context. If we go far back into the bronze age (into the period in which the heroic and Homeric epics are set), we hit the civilization of Mycenae. I won't delve deeply on the subject of Mycenae for this response (it's mainly important to be aware of its existence and then collapse), but what is important is its collapse, because with the fall of Mycenae, Greece fell into a kind of dark age. They still remained in the older settlements (many of the cities of Classical Greece long predate Classical Greece and stretch further backward into time), but in many respects, Greek culture and civilization largely had to restart. By the time we reach the eighth century, Greece begins to re-emerge. The dominant political organization of these early Polises was oligarchic.
These oligarchies were politically dominated by the wealthy, but they were also unstable. Indeed, many of these oligarchies were at some point or another (even if only temporarily) overthrown by tyrants, individual leaders who were able to gain enough of a following and support to overthrow their fellow elites and institute themselves as sole rulers of the city. In addition, population growth and unequal wealth distribution were only further stirring the instability and divisions that were feeding into this conflict. It was in this context that Dracon and Solon emerged.
Dracon is mainly known for devising a very harsh legal code, highly punitive, in which the vast majority of offenses seemed to have been punished with death. Solon emerged afterwards and offered far-reaching economic, social, and political reform. Solon banned the traditions of debt slavery, making it illegal for any Athenian to enslave another Athenian citizen, while also instituting land redistribution. To prevent potential famine, he outlawed the exportation of food crops, and, in recognition of the limitations of the Athenian soil, he encouraged the cultivation of olives and grapes. Politically, he set up a tiered class system based on wealth, allowed all citizens the opportunity to serve on juries, and offered all citizens the right to try lawsuits before those juries. In addition, he created the Council of 400, which also drew from all classes. As historians Ian Morris and Barry B. Powell suggest, "Solon wanted to strike a balance between rich and poor and to restrain elite feuding" (The Greeks, 208).
Citation Note: This answer was written with the aid of the following reference: Ian Morris & Barry B. Powell, The Greeks: History, Culture, and Society, Pearson Education, Inc: Upper Saddle River, New Jersey: 2006). The information concerning Dracon (this is the spelling chosen by Powell & Morris) and Solon, as well as the economic & political instability which necessitated such reforms, was found in pages 204-209. In addition, for further information concerning the dark age, see chapter 5 (pp. 72-92), and for further information concerning Classical oligarchies and tyrannies, see chapter 8 (148-170).