This exceedingly simple projection preserves the area of continents. That is the most important property when glancing at a map of the whole Earth. The first instinct we have is to compare the size of geographic features of the planet. While things are squeezed and amorph near the poles, shapes are accurate near the Equator.
This one preserves angles, which means that shapes are accurate. However, everything near the poles is exceedingly inflated. So much so that the poles are actually infinite and cannot be represented.
Notice how meridians (vertical) and parallels (horizontal) form squares? This projection maintains heights. A distance measured vertically corresponds to the same number of kilometers everywhere on the map. Of course, a horizontal distance is only the same along the parallel: as the ruler goes closer to the poles, that distance represents a smaller and smaller amount of kilometers. (Also, this is a good middle ground of pole squishiness between Lambert Cylindrical and Mercator.)
This one is a complement to the Equirectangular. It maintains widths. A horizontal ruler measuring a distance will correspond to the same number of kilometers throughout the map! Since the default rendering of this projection makes the map converge to a single dot at the poles, I favour this interrupted projection (or "gore map"), which does not distort shapes quite as much. As a bonus, this projection also preserves areas, although it is arguably not as readable as the Lambert Cylindrical.
We have a lot of maps with many incompatible properties. This projection gives an approximation of all desirable properties in a map. Hence, the name, German for "triple": it tries to preserve areas, angles and distances.