Some portions of their book should be accessible without too much background. Take a look at their sections on additive geometry, graph-theoretic methods, and algebraic methods, for example. For the bulk of the book, though, knowing some probability theory will make a big difference.
A recent book that I like and you might find more accessible is Alfred Geroldinger-Imre Z. Ruzsa, "Combinatorial Number Theory and Additive Group Theory", Birkhäuser, 2009.
From their Foreword:
This book collects the material delivered in the 2008 edition of the DocCourse in Combinatorics and Geometry which was devoted to the topic of Additive Combinatorics.
The two ﬁrst parts, which form the bulk of the volume, contain the two main advanced courses, Additive Group Theory and Non-unique Factorizations, by Alfred Geroldinger, and Sumsets and Structure, by Imre Z. Ruzsa.
The ﬁrst part focusses on the interplay between zero-sum problems, arising from the Erdős–Ginzburg–Ziv theorem, and nonuniqueness of factorizations in monoids and integral domains.
The second part deals with structural set addition. It aims at describing the structure of sets in a commutative group from the knowledge of some properties of its sumset.
The third part of the volume collects some of the seminars which accompanied the main courses and covers several aspects of contemporary methods and problems in Additive Combinatorics.
I would recommend that you work through the second part, and see how you find the material. It should be accessible.
You may also want to take a look at Ben Green's notes on the structure theory of Set Addition.
Let me add: If you are mainly interested in classical additive combinatorics, as it applies to the natural numbers, then I strongly recommend Melvyn Nathanson, "Additive number theory. Vol II: Inverse problems and the geometry of sumsets", Springer, GTM 165, 1996.