The book is itself representative of the research product of the program. The code that they wrote actually is mostly in Coq, to my knowledge, and certainly the development that accompanies the book was written in Coq.
Homotopy Type Theory itself essentially constitutes Martin-Löf, together with the univalence axiom, which essentially states that equivalent types are equal. The notion of equivalence comes from a view that establishes a connection between a synthetic homotopy theory and Martin-Löf type theory, hence the name. This axiom gives way to a powerful type theory that the book shows is capable of reasoning about a lot of important existing foundational mathematics.
In fact, part 2 of the book comprises developments of a more traditional, topologically-motivated version of homotopy theory, set theory, category theory, and a nice constructive presentation of the real numbers. I have not explored the Coq code that much yet, but my understanding is that all of these presentations are accompanied by corresponding Coq source. Since we can formulate all of these important foundations entirely within homotopy type theory, the hypothesis that it might be an important foundation for mathematics going forward makes sense. The idea is that mathematicians will be able to extend these seminal developments towards more results in fields like algebra, topology, and analysis.
Inasmuch as homotopy type theory pertains to being able to do mathematics in Coq via the Curry-Howard correspondence, you can think of homotopy type theory as corresponding to a more powerful instance of Curry-Howard. The most exciting contribution of this work is really that it gives mathematicians language for speaking about mathematics in terms of type theory, and the idea is that this should enable on the one hand, more mathematics being accompanied by mechanized proofs in provers like Coq, as well as computer proofs that correspond more closely to their plain English equivalents. Anyone who has used Coq will tell you that the proofs that you produce often don't end up looking much like equivalent plain-English proofs at all.