How did the pigs use Boxer's death to get the animals to work harder?
The animals are already working hard at the time of Boxer's death, but the pigs use his memory to try to reinforce an ethic of labor and obedience. While it's clear that the pigs sold Boxer to the glue factory and spent the earnings for a case of whiskey and a drunken revel, they make up a story that Squealer repeats. In this story, Boxer died in the hospital, having received excellent care. Napoleon purchased expensive medicines for Boxer, not for a moment thinking of the cost. The animals believe this story and are relieved to think that Boxer died happy.
Napoleon himself gives a short speech, which he ends with Boxer's two favorite maxims: "I will work harder" and "Comrade Napoleon is always right." Napoleon advises the animals that they should adopt these maxims for themselves. The pigs, knowing that Boxer was admired and influential among the animals for his hard work, sincerity, and dedication to Animalism, are trying to use his reputation to influence the animals.
The novel never says that this tactic is effective, but we do know that the animals toil in hardship at the end of the book, often hungry and cold, living in spartan conditions and no longer expecting the dreams of the rebellion, which most can't even remember, to come true.
The pigs used Boxer's death as a means for getting the animals to work harder by using his life as an example for the degree of work the pigs expected from the animals. They used Boxer's slogans for his own work and wanted the animals to apply it to their own work. They tried to make it seem to the animals that they were honoring Boxer's memory by subscribing to his slogans. Napoleon himself addressed the animals at the close of chapter nine and he reminded the animals during his speech that Boxer's favorite slogans were, "'I will work harder' and 'Comrade Napoleon is always right'". He made it a point to tell the animals that it would be in their best interest to adopt these slogans as their own.