To offset things in time or space so they do not land together. Staggering spreads a load that would otherwise arrive at once.
Old Norse stakra (to push, to totter), a frequentative of staka (to punt, to push); the unsteady walk gave its name to any deliberate spreading-out that keeps a load from landing all at once.
Offsetting things in time so they do not arrive together. The unsteady walk the word first named is exactly what a staggered load avoids: everything landing on the same foot at once.