In the movie, the representatives themselves are sitting there fighting with each other. Not their aides or interns, but them. Now I never see video of senators or representatives arguing back and forth. Supposedly it is all done on committees during the work day.
How did this evolve over time? Why did it evolve like this?
Congressional staffer here. The only time you would see all the members of the House on the floor of the House is when they are voting on a bill, and you'll also see all the members of both houses when there is someone speaking before a joint session of Congress (like the state of the union or a foreign dignitary). When there is a floor debate, usually the only members there are those who have something to say, and otherwise they will be in their office working on other issues.
When Lincoln was a member of the House, most of the members worked at their desks in the old House Chamber (now the National Statuary Hall). In 1864, the current House Chamber was built, and members would generally work in committee rooms.
In 1908 when the House Office Building (now known as the Cannon House Office Building) was built, each member had their own office to work in.
In 1933, the Longworth House Office Building and in 1965 the Rayburn House Office Building opened. When these opened the offices got larger to the point where in the Cannon House Office Building each office today is three of the original offices merged together.
As far as why it got like this, as the government grew over time, the members had more responsibilities and needed more office space.
Source: there's an exhibit in the Capitol Visitor Center that has a model of the Capitol Complex at each stage of expansion and has information that goes into detail about what the Capitol was like at that time.