In short, a staggered press (also called “piano keys”, “zurashi oshi”, “drumming”, “slide input”, and a lot of other names) is a specific way to mash across a few buttons.
It is also a type of chain, if you would prefer to think of it that way.
Staggered press is useful when timing is highly severe, but the actual requirement is that one of a number of buttons be pressed. For example, you may have a very tightly-timed cancel in a combo, or even tougher, a “link” (no cancel). This is highly apparent among Street Fighter III players, because used in tandem with negative edge across all three punch or kick buttons, you get a maximum of six chances to hit the input instead of the two chances with one button – which is excellent if you know you need it to register in the correct two-frame window, for example.
A different use is found in 3D-type games, where it is necessary for some command inputs.