One way is to break the DNA down piece by piece one or a few bases at a time but this is not a very efficient way.
Most sequencing today is done using enzymes that make copies of the DNA one bases at a time.
When the plates are separated a sheet of film is placed on the plastic gel. When developed, there are bands that can be read to give the sequence. Many laboratories today use fluorescent labeling of the DNA and machines can detect a specific bases by there emission spectrum. A molecular biology text book will supply you a nice diagram explaining this in a much simpler way. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.