Loving your enemy is very difficult to do. Most of the things Jesus said in the "Sermon on the Mount" (that particular part of Matthew) are ideals, such as "do not worry" (we will inevitably worry about things not worth worrying about). Another thing to point out is that Jesus said any man who looks at a woman with lust is committing adultery. Although the context of the time is worth mentioning (most adults were married so if you saw a woman it was likely that she was married), that's still a very high standard.
Loving your enemies illustrates that idea as well. For the Jewish audience, they were used to "an eye for an eye"--in Mosaic tribal law you murdered someone the family of the victim was allowed to get revenge by killing you, unless you fled to a specific shelter city. For us Mosaic laws sound harsh, but when Jesus says that not only must you not even look at a woman lusefully (same thing as illicit sex), you can't even desire revenge or harbor hatred for someone, even if you have a very good reason (people tend to become our enemies for a reason).
The point Jesus was making was not "I'm going to make the law even harder for you to follow" but "the law is deliberately impossible. You can do everything right but if your thoughts are impure or against the law, then it's the same thing as breaking the law".
The mosaic law placed an emphasis on actions--it's hard to have someone executed for "thoughtcrime" in the real world. The PHarisees emphasized this with their reliance on actions (or prohibitions). They added to the law, thinking it made them more pious.
But Jesus was showing that the spirit of the law was more important. Even Deuteronomy encourages having the commandments in your "heart" (emotional being), rather than just in your head. It's in one of the Psalms as well.
Jesus's sermon was radical for the time. He was telling the Jewish people to disregard the specifics of the Old Testament law in favor of faith and willingness to not sin. Remember that many people Jesus interacted with were repentant sinners or people who should have been hated by their society for their supposed disregard of the law--prostitutes, tax collectors (associated with greed and obedience to a foreign power, Rome) and fishermen (uneducated rednecks of the day).
To get back to loving your enemies, well, the point of verses like it is that as a sinner, you we'll never be able to not to. I've struggled with hating people for a while now, usually anybody that isn't like me--mainly because I suspect that they would hate me anyway even if I was nice to them. But the verse encourages getting over that, knowing you will fail at times, as long as you try to let go of hating people, even if they deserve it. And most importantly, you're not doing it for any pragmatic reason--not because it's socially accepted or "right", but because you are choosing to resist the temptation and want to let the hatred go, even if it seems to fuel you. If someone's Christian faith compels them to stop hating their enemies, it shows that they are wanting to follow the ideals of a higher power greater than themselves, and not just society, which it would have been ffor the Jewish people (the difference being that their society encouraged hatred for good reasons whereas we do not, thanks in part to lingering Judeo-Christian ethics that encourage forgiveness and reconciliation).