I'll give it a shot.
bon- = good.
voli = to wish or to want.
bonvoli means to want what's good [for someone]. This is reinforced by the noun form, bonvolo, which means affection or goodwill.
Putting it together, "Bonvolu [verb]" should translate roughly as, "[I insist that] you want it to be good to [verb]." Which is a mouthful to say, but pretty much what we mean by, "Please."
(It's a slightly different perspective, though. To please is to make someone happy, while bonvoli is more about you being happy to do something.)