I have no background in string theory but have a reasonable exposure to quantum field theory including the quantization of gauge theories. In simple terms what is this Gauge-Gravity duality all about and its connection with AdS/CFT correspondence mentioned in this context?
First, of all as was pointed very keenly by @Antoine the book is pretty much self-contained and guides you very well.
Secondly, in very pop terms and without using formalism let me try to answer the question:
"In simple terms what is this Gauge-Gravity duality all about and its connection with AdS/CFT correspondence mentioned in this context?"
The holographic principle as an idea was proposed originally by 't Hooft as an attempt to resolve the black hole information paradox. Later it was given a string theory realisation by Susskind.
Further down the road Maldacena realised that there can be a well defined example of this holographic principle and this is what he formulated. Take TypeII on AdS$_5 \times$ S$^5$ and this is shown to be dual to a particular superconformal field theory in the boundary. We now know that there are ways to dualize TypeIIA, TypeIIB in 10-dimensions, as well as their $11$-dimensional lifts.
What is more to that is the following. People, due to the compelling evidence, have put forth the notion of gauge/gravity duality. This would mean that you do not necessarily need to start with some kind of superstring/supergravity theory to have a dual, but rather a gravity theory in your bulk. What this is suggesting is a stronger statement, particularly instead of talking about holographic duals in the context of a superstring theory, the claim is that there exists a dual gauge theory for a gravitational theory in the bulk. An example to that is AdS/QCD, Dynamic AdS/QCD which is a modification of the former and there are many form in the literature.
I hope this shed some light to your question. If not, let me know what else I can try to answer. If you don't like the answer, I will delete.