What is an example of casework?
Having worked on many cases on behalf of the constituents of the members of Congress for whom I worked on Capitol Hill, I can provide many examples of casework. My first two employers served on the House and Senate Armed Services Committees and both had numerous military installations in their districts or state. Around any military installation is a network of contractors who perform much of the day-to-day work of keeping those installations operating. Inevitably, some of those contractors, who might perform any number of services for the military, from providing food services to designing and building weaponry, will complain to the congressman or senator about delayed payments for their services. The congressman or senator's legislative assistant or caseworker (legislative assistants work in the congressman's Washington, D.C. office and caseworkers are located in the district or state and are, consequently, more immediately accessible to the public they serve) will open a file on this case, which will involve contacting the relevant government agency (in this case, the Department of Defense or the Department of the Army, Navy, or Air Force, depending upon the individual military contract or installation) with a request to have the constituent's complaint reviewed. Often, the caseworker will request a briefing by the relevant government agency. There follows a series of back-and-forth communications until the case is resolved.
Another example of casework, and a very common one, is when retired individuals do not receive their Social Security check. They contact their congressman's office, which in turn contacts the Social Security Administration on the constituent's behalf to request that the missing or delinquent check be reissued.
Caseworkers exist to represent the interests of their employer's constituents in all matters involving federal agencies.
The term “casework” is most commonly used as a term in political science. It refers to things that representatives do for their constituents. I have therefore moved this question to the Political Science group and will assume that you are asking about casework in this sense of the word.
Constituents often have trouble with government agencies. When this happens, they sometimes will go to their representative (or, more likely, that person’s staff) and ask for help. For example, I had a student who was in the US illegally. After he was in college, he was told that he needed to go back to Mexico and then fill out some forms to get back into the US. Various people connected with the high school that he had attended contacted our congressman. They got his help in navigating the bureaucracy. He presumably made calls to ensure that my ex-student’s case would not simply get lost in the bureaucracy. This is an example of the things that members of Congress do for constituents. These services are called casework.