What is the main thesis of On the Incarnation of the Word by Athanasius?

The simplest answer to your question is that the main thesis is given in the title of the work itself:  St. Athanasius is writing "On the Incarnation of the Word."  That is, he wrote this treatise to explain his belief and his religious faith, in a way that he expects will win over non-believers through logical analysis of some historical facts. 

Athanasius lived and wrote in the 4th century, when the Christian church was still very new, relative to today, and defending the meaning of faith was a real issue.  Writers like Athanasius saw the need to defend their beliefs. 

In the opening section of the treatise, Athanasius sets out four primary reasons for his work:

1) "in order that, all the more for the seeming low estate of the Word, your piety toward Him may be increased and multiplied"  [Athanasius is writing to someone named Marcarius, and says he wants to help increase his level of faith]

2) to "win over" those who "mock" and "disbelieve" to the Christian faith;

3) to "recall what has been previously said" - that is, to review earlier writers and their arguments

4) "so it may be duly perceived that the renewal of creation has been the work of the self-same Word that made it at the beginning" - in other words, to provide compelling argument that connects Christ and the resurrection to the same God who made the universe as related in Genesis.  In this way, Athanasius is speaking to Jews and Greeks who have faith in the Old Testament, but have not put their faith in the Christ of the New Testament. 

Athanasius then works his argument through 57 numbered sections to expand on these primary points of his thesis.  C.S. Lewis, author and Christian writer, has broken the work into 7 major topics:  1) Creation and the Fall; (2) The Divine Dilemma and Its Solution in the Incarnation; (3) the Death of Christ; (4) the Resurrection; (5) Refutation of the Jews; (6) Refutation of the Gentiles; (7) Conclusion.  These topics follow the outline of the four theses that Athanasius sets forth in his introduction.

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