You will need to purchase the baler for $5,000 to make progress towards this. After you purchase it, employees will start to insert boxes into it instead of the dumpsters. Once enough boxes have been put into the baler, it will spit out onto the ground a bale. You can then pick up the bale and take them to the recycler dumpster across the street. You can check your progress on the Manager Blackboard under the Statistics section. Once "Cardboard Bales Recycled" hits 40 you are good.
This achievement will unlock once you close your store for the night after you have recycled your 40th cardboard bale.