Center points have several uses.
See NIST material on center points
Adding center points to a DOE will help screen the design space for curvature.
Note that center points will only show the presence of curvature in the response. They will not quantify the amount of curvature, nor will they tell the experimenter which process variable(s) is (are) the cause of the curvature in the response.
To quantify the amount of curvature, or to know which processes variable is influencing the curvature, a 3-level design is needed.
Center points can be used to monitor the process for stability during the DOE.
DOE does not work well for processes that are erratic or otherwise poorly behaved. If possible, an SPC study should be used to demonstrate that the process is in a state of statistical control (stationary process mean and variance). If an SPC study is not possible, then using center points at the beginning, middle, and end of the DOE treatments can provide evidence of a stable process.
Center points are useful in Central Composite Designs (CCD).