A huge emphasis in the Athenian constitution was the expectation that citizens would be responsible with regards to politics and law.

One example would be that of service in the Boule, a sort of congressional body. It was basically a council of 500 men that was chosen by pure lot from every man within each of the 10 tribes (Each tribe providing 50 men). A similar idea is given off, in that it is expected that every male citizen should be able to show competence in state affairs.

Another case is the practice of the red rope in getting people to attend the Ekklesia, or the main citizen assembly. What would happen is that on meeting days of the Ekklesia a group of slaves would go to the Agora with red ropes so that those with the red stains that did not attend the meetings would be fined. And this was obviously to get more people to attend as every citizen was expected to participate.

So just as these bodies assumed competence from all citizens, that was the logic for the judicial system, in that citizens were supposed to know the ins and outs of law enough to defend themselves in court.

Source: Constitution of the Athenians, Aristotle

Edit: clarification, I'm talking of ancient Athens only, I can't really say for Greece as a whole