This was one of my earliest excursions into game development. Done for a now defunct project, I was responsible for the camera motions you see.
The initial goal was to realistically simulate a soldier's movements, thus giving the player a more 'immersed' feeling. This was solved by using various equations and timings to manipulate the camera angles/origins.
In retrospect, wow! What a complicated solution! Today, people will attach simple animations to a camera object. Sometimes, it will be a combination of the two, allowing variables to hook into the camera and manipulate it.