Scientists aren't entirely sure why we kept this hair, but they have a pretty good guess. We know that eyebrows help keep moisture out of our eyes when we sweat or walk around in the rain. The arch shape diverts the rain or sweat around to the sides of our face, keeping our eyes relatively dry. The most obvious advantage of this is that it lets us see clearly when we're sweating a lot or out in the rain. Without eyebrows, getting around in these conditions is a little more difficult. The shape of your brow itself diverts a certain amount of moisture, but eyebrows make a significant difference in your ability to see. Diverting the sweat away is also good because the salt in sweat irritates the eyes, making them sting a little.
There are a number of ways these qualities might have helped early man survive. Being able to see more clearly in the rain could certainly help you find shelter, and there are several circumstances when keeping the sweat out of your eyes could save your life. If you were trying to outrun a predator, for example, it's a good bet there would be a lot sweat running down your face. If all that sweat flowed right down into your eyes, you wouldn't be able to see that well, and your eyes would be irritated, which would certainly impair your ability to escape! Because of this slight survival advantage, nature would most likely select for humans with eyebrows over humans without eyebrows.