But watching the line is more than an aesthetic experience. It is also a way to judge what I'm doing. By watching the line, I can determine whether I'm making a good cast or bad. Given a little information, you can use this same diagnostic tool.
Let's begin with the understanding that the line doesn't just jump off the rod tip in some random manner. No, in fly-casting, the line actually unrolls, forming a candy-cane shape that we call a loop.
Properly executed, the loop looks something like this:

Or it might appear more pointed, like this:

Both shapes represent good casts. Note that there is not a big distance between the fly line's upper and lower legs. We say these loops are tight, or narrow. You might also say we are drawing a straight line to the target with the rod tip, focusing our cast. The closer we get to "drawing" a perfectly straight line, the tighter the loop becomes. This is an efficient cast. We're not wasting a lot of energy.
What kind of loop wastes energy? A wide loop, which looks more like this:

Notice that the wide loop has lots of space between the upper and lower legs, and the "face" of the loop is much more blunt. Most of your energy is being used to suspend line in the air; very little is being used to carry the fly to your target. The result is that you end up working much harder than necessary. Meanwhile, both distance and accuracy suffer.
How do you correct a wide loop?
You are probably driving the rod tip through a very wide arc, much like the windshield wiper on a car, and that wide arc is literally pulling your loop apart. Resist flexing your wrist when you cast, and stop the rod tip higher, well above your head, on both the back cast and forward cast.
That leaves us with the tailing loop, which looks like this:

A tailing loop means the line is actually crossing itself and tangling. If you've ever noticed a mysterious knot in your leader, known as a wind knot, that means you've thrown a tailing loop, even if you didn't see a tailing loop.
Figuring out what causes a tailing loop can be a bit complicated, because multiple faults are possible. But in my experience, most tails can be corrected with a few simple measures.
Focus on a stroke that accelerates smoothly. Don't rush it, and don't force it. Try lengthening your stroke, and resist any temptation to power your way out of the problem. You might find that casting with less power eliminates the tail.