Flow is where classical Pilates beats modern Pilates hands down. There is something graceful and dance like when you perform an entire Mat class in the classical order. That said, modern, non-classical styles can flow very well also. This is the one place where I feel like successfully executing the principal is 90% up to the teacher and only 10% up to the student.
Flow is simply the flowing from one exercise to the next. The body is in continuous, purposeful, controlled and mindful movement. Your Pilates experience should be like a dance.
Flow happens when the other principals happen. You cannot truly have it if any of the other pieces are missing. If you are not a dancer (I'm not) achieving flow is one of the most satisfying parts of Pilates because you feel elegant while you exercise.