Why did the Athenians feel it was important for citizens to take part in the government?

They tried kings, then switched to oligarchs (aristocrats) to lead them. The oligarchs abused power for their own benefit, and ran a society where farmers who could not repay debts after a bad season were sold into foreign slavery to pay out the debt. As revolution was looming the aristocrat Solon was appointed dictator to solve the problem. He did some good, but the aristocrats re-asserted control, so the people appointed a tyrant to establish a rule which was fair to all. This experiment, after initial success, began to fail and so a new form of government was trialled - democracy where the people (adult male citizens) met in assembly and voted on political matters. It worked for a period, but inevitably populist democratic leader started twisting things to the benefit of themselves and their cronies, and led the city into disastrous war, so the city reverted eventually to trying versions of the earlier forms of government.