Above the threshold of the oracle at Delphi were three inscriptions which are characteristically Greek. The one to focus on here would be "nothing in excess". The Greeks believed that people should live life on a fine line where they sought to achieve personal excellence while at the same time remembering their place within the universe. They should strive for the best that they can, but should not strive to become like the God's themselves. This theme is common throughout mythology and when one tries to exceed their abilities and rise up to the level of the God's, they are always cast down. Examples of this would be Niobe, Bellerophon, Tantalus, and many many more. Icarus is a character who goes not only against the words of his father, but also tries to ascend to heaven and exceed what is humanly possible. As a result of this, the wax which holds his wings together melts and he eventually meets his doom. He tries to exceed his own limits and stand against authority and as a result is struck down.

I admit that I may not be right about this, but the story of Icarus is one of those myths for which we don't actually have a particularly compelling Greek account of. It is described in Apollodorus, a second century encyclopedia, and alluded to frequently, but the version you are probably familiar with is that of the Roman poet Ovid. In this telling, it has a very clear moral lesson on hubris. I can't find a translation that captures the essence, so here is a quick and dirty one:

8:202 instruit et natum 'medio' que 'ut limite curras,

8:203 Icare,' ait 'moneo, ne, si demissior ibis,

8:204 unda gravet pennas, si celsior, ignis adurat:

8:205 inter utrumque vola.

"You must hasten on the middle path, Icarus" he said "I warn you, if you go lower, the sea may weigh down the feathers, if higher, the fire may scorch them: fly between each side"

So, pretty clear I think.