Your question has nothing to do with Newton's third law.
You are having trouble identifying forces. That's not unusual. It takes some time and practice. A good book would help. Evidently your book is not very good at all, because the sentence you quote is practically unintelligible. Please don't waste your energy trying to make sense of it.
Wikipedia has fairly good statements of Newton's laws, although I think the statement of the second law, while correct, could be even better.
First you have to define your system. You have implicitly chosen your system to be the rope. Then you have to find the forces on the system. In Newtonian mechanics, a force is either a non-contact force (gravity, electrostatic, magnetic) or a contact force (everything else: normal forces, tension, the push of your finger, friction...).
In your question, we are implicitly neglecting all of the non-contact forces. That makes things a little easier. We are looking for contact forces. These only occur when the system is in contact with something it its environment. Here, they occur where the forces are applied to the rope, say, by your hand pulling on it. So there are two forces on the rope, one on each end.
A force that is a "pull" is usually called a tension force. So you have two tension forces, and they happen to be applied in opposite directions. Now, the rope is at rest, is it not? It's acceleration is zero. Newton's second law tells us that the net force, the algebraic sum of the two forces, is zero. Thus, the forces must be of equal magnitude and opposite direction. This has nothing to do with Newton's third law, although it shares some of the words used in the statement of Newton's third law.
Note that if the rope were accelerating, then the two tension forces would be in opposite directions, but unequal magnitude.
So here's what we have: two forces on the rope. Each one is applying a tension force on the rope. Those two forces are equal in magnitude, and opposite in sense (direction).
The only introductory book that I've seen that spells all of this out clearly is out of print.