These are two meanings of the word. The second one applies to your example about the board. Interestingly though, your example makes use of both definitions.
- To place on or as if on alternating sides of a center line; set in a zigzag row or rows: theater seats that were staggered for clear viewing.
- To arrange in alternating or overlapping time periods: staggered the nurses' shifts.
A hostile bidder could be seen as bowling ball and the board members as pins.
If they are all in a straight line (all board members coming up for re-election every year instead of every three years on a rolling, or staggered*, basis) the entire board can be thrown out (knocked down) as a group in a proxy fight initiated by the hostile party.
If you stagger their re-election terms, they make a much harder target!
*This should not be confused with a staggering board. That would be the same group of people, near midnight, during the Christmas party.