Each level has its own requirement.
Level 13: Use only two Monkeycopters and no grabbers. If you are focussing on this achievement, it isn’t hard. Just run through the level quickly, ignoring any other objective and be careful not to hit your monkeycopter on anything (including the edge of the ring of fire). Don’t switch copters until you absolutely need and mop up the other objectives on a separate playthrough.
Level 14: Use only three monkey copters and no grabbers. The same advice from level 13 applies, avoid everything and just focus on passing the level.
Level 15: Use only two Monkey Copters and two grabbers. Same advice as level 13, though this level is a little trickier.