As usual, Father Stephen from Glory to God for All Things, says it much better than I could.
Why did Jesus tell us to love our enemies? Because that is who God is.
God is not the divine boss-from-hell. He does not exercise a power trip over His creation from his lofty place in the sky. Jesus did not come to burden us with the minutia of impossible-to-follow commandments--Love is the commandment that covers all others. Jesus came to show us God and to lead us to God. That is the sole purpose of this life.
The world had gone so far astray from God that He sent His son, God Himself as the Church teaches, to show us what Love is. Love is not taking revenge. Love is dying to one's self for others. Love is mourning the sins of one's enemies rather than plotting revenge.
The kind of God who asks us to love our enemies is the God who, in His infinite mercy, loves us despite our own sin (literally "missing the mark"). He constantly calls us back, searching for us in the garden when we have hidden ourselves in shame.
Without love, the world makes no sense. Without love, nothing works. Love is all.