Apollo's twin sister, the goddess Artemis was one fierce huntress. Smart, quick with a bow, and fiercely independent, she loved running through the forest, searching for prey. She also had a posse of protective nymphs who served as her loyal back-up crew.
Now Artemis didn't take very kindly to suitors. See, she wanted to stay independent and feared that relationships with men would decrease her power and freedom. Fair enough, we guess. After all, in ancient Greece, women were often beholden to their husbands. So what better way to stay in control of one's own fate than to avoid men altogether?
Artemis was so committed to this no boyfriends thing that she remained completely chaste. Publicly, she always presented herself in hunting gear in order to seem strong, confident, and in-control. So when Actaeon sees her naked, it's a big deal, because men never get to see Artemis naked. That means that by peeping on her when she's nude, Actaeon is (1) barging into her sacred female space, which feels like a threat to her chastity; and (2) diminishing her status and power, by seeing her vulnerable, nude, and not totally in control.
To make matters worse, modesty was an important part of Greek culture. Actaeon seeing Artemis without her clothes on is a huge breach of modesty and respectability, and this combination of shame (at a loss of modesty) and anger (at the loss of her power) is a deadly one.
Unfortunately, this myth is sometimes used to portray Artemis as an emotionally unhinged female. Ovid, in particular, writes the story from this angle in his Metamorphoses. Not cool, Ovid.
But what Ovid might see as emotional, we can also read as just plain fierce. Artemis's quick wits and even quicker temper meant that she had a vindictive sense of justice. By changing him into a deer (instead of shooting him with her bow and arrow), Artemis made Actaeon suffer and taught him a serious lesson about power. He dared to take away her status by barging in on her sacred space, so she made him totally powerless by transforming him into a scared stag.
It's unclear whether Artemis intended for Actaeon's dogs to eventually kill him. What do you think? Was that part of her evil master plan, or just an unintended bonus?
P.S. For much, much more on Artemis, check out Shmoop's files.