What you describe is often called the "focus and recompose" technique. It was first used back in the days of manual focus with a single focusing aid, usually a split prism, in the center of the camera's viewfinder.

Autofocus systems focus the lens at a specific distance from the camera, not at a set of coordinates in Cartesian space. As long as the distance from the camera to the subject is the same when the camera was pointed directly at it as it is when the camera is moved to reframe the subject, then the subject will still be in focus.

One thing to keep in mind is that when panning the camera to change the composition, the camera should be rotated around the optical center of the lens, also known as the "no parallax point" and sometimes incorrectly referred to as the "nodal point."

Most folks who aren't actively thinking about it tend to rotate the camera around the center of their own body, rather than rotating their body around the optical center of the lens. This changes the optical distance between the camera and the subject. Sometimes the difference is less than the depth of field resulting from the selected aperture, focal length, shooting distance, and intended display size and the subject still appears to be in focus. With wider apertures and short subject distances, though, the depth of field is fairly shallow and it is often the case that the subject will no longer appear to be acceptably sharp when using the "focus and recompose" technique.

Related: What is the advantage of a lens with a curved focal plane?