Your main system is the Salvage, this is the beam that fires to harvest your blocks.
The secondary system is the Cannon, these blocks don't fire, they just give a support effect to your main system.
You need Cannon blocks linked to the Cannon computer which is linked into the Salvage computer in order to "power" the secondary effect of lowering your cooldown time for the salvage beams. The more cannon blocks you link in, the more it reduces the cooldown. Think of it like a throttle. If you only want a little cooldown reduction, you only link in a few cannon blocks. If you want a lot of cooldown, you link in a lot of cannon blocks.
You checkerboard the Salvage beams in long single rows because any blocks that touch each other count as being part of one beam. So a 3x3x1 or a 9x1x1 arrangement each produce a single beam of equal strength. By checkerboarding them, you get lots and lots of beams for a very wide area. Its the difference between having a .22 and a shotgun.
While its not important where or how you put in the slaved cannon blocks (they don't fire, so only the number you have linked in matters), the fact you have empty space in your checkerboarded salvage lines makes that a great place to stick your cannon blocks too.