I have now read in several different books which relate the adventures of world travelers the same observation about the sunset at the equator. They state that at the equator "darkness falls almost instantly after sunset, there is no twilight". I ignored it the first few times I saw it, but after reading it again, and again, I began to wonder if it is true, and if so why? I can't figure out why twilight should shorter or longer anywhere in particular. Is is true? Why?
This is a very good question. The reason is a little complicated to understand, though. The simple answer is that at low latitudes, the sun sets perpendicular to the horizon, while at higher latitudes, the sun can set at a more oblique angle, allowing it to remain close to the horizon after sunset for a longer period of time.
The sun rises and sets because of the Earth's rotation. It's easy to understand how things move in the sky due to the rotation by looking at a long-exposure photo taken at night. Each of the circular streaks there is a star, and over the hour or so that the photo was taken, each star appeared to circulate around the North Star, Polaris, seen in the center of the pattern.
You can see from the picture that stars closer to Polaris than the horizon never rise nor set, but continually circle the pole star in the sky. Stars farther from the pole star do rise and set, as the circles on which they travel cut below the horizon. The same idea goes for any object in the sky, including the sun and the moon.
One more thing you need to know about these rotation patterns is this: Polaris sits directly above the North Pole. If you were standing at the North Pole, you would see Polaris directly overhead. All the objects in the sky would simply circulate around that point, and nothing stationary in the sky would rise nor set. This makes sense if you think about standing at the pole, while the Earth's rotation simply spins you around in place. You end up looking in different directions, but nothing enters or exits the sky above you. The closer to the north pole you are, the higher Polaris is in the sky. At the Equator, Polaris is on the horizon.
This means that at the north pole, the sun doesn't rise or set at all due to the Earth's rotation. The sun does move through the sky (from one of those circles on the film to another) on a yearly cycle, though, and this is why there is light at the north pole for 6 months and night for 6 months. Thus, above the arctic circle (or below the antarctic circle), at the right time of the year, you can see the midnight sun-- a sun which doesn't set all day. At that time of year, the sun is a circumpolar object, sitting on one of those star-trails in the picture that never descends below the horizon.
Now imagine that you are pretty far north, but not above the arctic circle. Polaris is very high in the sky, and as the Earth turns, the sun may be on one of those circles you see in the photo which just grazes the horizon. The sun sets, but sets at an angle. It stays close to the horizon for a long time, making twilight linger.
The further south you travel, the lower Polaris will appear in the sky, and the further and quicker the sun will dip below the horizon at night. Eventually, once you get to the tropics, Polaris is on the horizon, and the sun sets directly down below the horizon. Since it has very little movement horizontally with respect to the horizon, it quickly leaves the horizon behind after sunset, making for a very quick twilight. Then, as you proceed southward from the equator, the effect is reversed. Twilights become longer and longer, you see the South Celestial Pole higher and higher in the sky, more objects become circumpolar (i.e. never rising or setting), until eventually you're at the south pole, and the South Celestial Pole is directly over your head, and once again everything is circumpolar, and nothing rises or sets due to Earth's rotation.
I should say one more thing here: there is nothing inherently special about the star Polaris. It just happens to be very near the North Celestial Pole, the geometrical point directly above the Earth's north pole. There is also no corresponding bright south pole star.
I hope this gives you a better understanding of sunrises and sunsets!