He is a man of courage who does not run away...
What did Socrates mean? How does this relate to him
What Socrates meant by this quote was that a courage, honorable persons would not bail at the sight of death, but would face the challenge like a honorable person. In Socrates' day this could mean war, probably what he was referring to. Socrates fought in many sieges of Athens defending his home. That takes courage, most likely the courage he talks about in this quote.
How does this quote relate to Ancient Greek virtues?
Socrates' quote relates well to what was going on in Ancient Greece. Athens was a direct democracy, the people were the power. It was up to them to stop incoming invaders, this meant everyone needed to man up and fight. There were always battles between city-states and it was very frowned upon to chicken out of battle. Even Greek religion favored those dying in battle, with the best afterlife, then those leaving his peers in battle.
"Socrates" Google Images 20 September 2015
"Athen's battle" Google Images 20 September 2015