The word, "evil," is used in this phrase to mean the troublesome, annoying, problems of everyday life. In the Matthew version, it reads, sufficient unto the day is the evil thereof. In other words, every day brings enough problems that we don't need to waste our time worrying about the problems of tomorrow or the next day. We should live in the present and concern ourselves with today's issues. Within reason, we are to live life one day at a time.
Next, we should review how the text was changed for the Nephi version. It reads, sufficient is the day unto the evil thereof. Notice that the placement of the words, "is" and "unto" are switched. This approach is more optimistic. It focuses on the sufficiency of the day not the sufficiency of the evil. In other words, the Lord provides us with enough time and means to effectively deal with all of our problems. The key for us is to learn to live life without worrying about every pothole in the road.
This is such practical advice! Some have perfected worrying to a science. Others have fashioned it into an art form. Yet, worrying doesn't solve any problems. It just raises blood pressure, hardens arteries, and shortens lifespans. In the eternal scheme of things, the small things are not worth the sweat and tears we waste on them.
Brigham Young
"It's basic to realize that we don't run things. We are not in control. We can't make people do anything, and so we have nothing to lose. So don't get flustered and don't worry. Your Heavenly Father is in control." (Brother Brigham challenges the Saints, p. 462)
Harold B. Lee
"…the only day you have to worry about is today. There is nothing you can do about yesterday except repent. That means if you made mistakes yesterday, don't be making them today. Don't worry about tomorrow, because you may have no tomorrows. This is the masterpiece you ought to be thinking about today. And if you can always witness honestly that whatever you did, you did to the best of your ability, and next day try improvement on that, when your life's end comes, of you it can be said in truth, his was a successful life because he lived to the best that was in him. That's all the Lord expects of any one of His children. We are all born with different capacities, some to do one thing, some to do the other, and all He asks is that we do our best; and that's the measure by which we'll be judged when that time comes." (The Teachings of Harold B. Lee, p. 64-5)