I think he means should be "recover" from the soreness in his muscles after stretching before doing the next session, similar to if you have severe DOMS one would normally give an extra day of recovery before training that body part again.
My answer to the question would be, "it depends". If you are getting that much pain from stretching there is tons of microtrauma occurring at the muscle/tendon. BUT, you shouldn't be getting THAT much microtrauma if you are stretching properly. My guess is that you are either using improper technique on your stretches and/or applying way too much over-pressure and being too aggressive with each individual stretch. Also, if you aren't stretching when the blood supply is ample to the area and the muscles are not "warm", that can cause some of that discomfort as well. I think it was Myers that said that it can take upwards of 12 months with regular soft tissue manipulation (mobility drills, foam rolling/soft tissue work, stretching etc) to create significant and permanent plastic changes in myofascial restrictions so it isn't something to rush through.
Or, you could just be over analyzing a normal muscle soreness and have a lower pain threshold and don't need to worry about anything.