This expression might be considered a bit of a metaphor. We know we have organs in our body that do the work necessary to keep us alive. Our hearts pump blood, our stomachs process our food, and our lungs make sure all of the body gets oxygen. In the same way, the organs of government do the work of government to keep it alive and functioning.
In the United States, the three branches of government set up by the Constitution are the three primary organs of government, the legislative branch (Article I), the executive branch (Article II), and the judicial branch (Article III). The legislative branch does the work of making laws to govern the nation. The executive branch appoints cabinet members and members of the judiciary, acts as commander in chief of the military, and sets policy and tone for the nation. The judicial branch rules on cases involving federal statutes and constitutional issues, as well as a few other kinds of cases.
Other nations have somewhat different organs of government. For example, in the United Kingdom, there is a Parliament composed of a House of Lords and a House of Commons. A prime minister is elected by Parliament and has the power to make appointments, enter into treaties, and deploy the military. There is a judicial branch, and there is a monarch who is mostly a figurehead. Still other countries have somewhat different organs of government, as a theocracy will have a religious leader at its head and possibly an organ to enforce religious compliance. But all governments need organs of some sort or another. Even a dictatorship needs to have entities that carry out functions.