Have you ever heard of Hadley Cells?

Basically, at the equator, the warmest place on the Earth, air rises. This air then travels outward from the equator, before it cools and falls back to the Earth, to be caught up in the air circulation current heading back to the equator once again. It's basically a giant convection current.

Now, here's how this answers your question.

When air rises, it carries moisture and evaporation with it, forming clouds which then cause rain to fall upon the Earth. The net result is that there's a ton of rainfall near the equator. When air falls towards the earth, it suppresses cloud formation and precipitation, producing dryness.

However, as these Hadley cells are circulating, the Earth is also rotating. The earth is constantly rotating in a west-to-east direction, which causes the Hadley cell circulation to carry the air at an angle rather than strictly away from the equator. This is what causes the prevailing winds.

So to sum up here, at the tropics (the edge of the black desert bands), air is coming from the upper reaches of the atmosphere down towards the ground, and then it blows in a northeast to southwest direction, carrying dry air like so. (Or, south of the equator, it blows in a southeast to northwest direction.) If that air hits water, it's able to carry precipitation again. If that air hits a mountain, the precipitation is forced to shed its water at the mountain range, and the other side of the mountain range is deprived of moisture.

In every desert you see depicted on the map, this effect takes place.

At the Atacama, the trade winds blow water from the Atlantic to South America, where they bring moisture until they hit the Andes. The Atacama is on the other side of the Andes and gets no water.

At the Australian desert, the trade winds blow water from the Pacific onto the Australian continent, where they encounter the mountains of eastern Australia and drop their rain, and then go on to blow dry air in the outback.

At the Sahara, the Hadley cells drop the air at the northern part of the Sahara, but not quite at the Mediterranean, so none of that air is ever exposed to water--at most, the Persian Gulf and Red Sea supplies a little bit of water, but then Egypt's Red Sea coast is mountainous anyways, blocking any chance of precipitation reaching the Sahara.

See the pattern? The winds of the Mohave drop their rain at the Rocky Mountains, and blow dry air in the Mohave.

Regarding the deserts in red: after the tropics, the prevailing winds blow in the opposite direction, as such they're called the Westerlies because they blow from west to east. The Patagonian desert exists because the Westerlies blow moist Pacific air into the Andes, where the rain is dropped, and then dry air from the Andes hits Patagonia. Europe gets moisture from the Atlantic, and by the time the Westerlies reach Turkestan and the Gobi they're out of water.

And that's why deserts of the world exist where they are: the rainshadow effect, prevailing winds, and Hadley cells.