Some clarifications to the question are needed. First, you are referring to Section 16.1 of Serre's book (not 16.2), where he is formulating the main results in Brauer theory. These were originally derived (as in the 1962 Curtis-Reiner text) more concretely in terms of Brauer characters, but then recast in the language of Grothendieck groups. The result you are asking about is formally stated by Serre as Corollary 2 of Theorem 35 in that same section.
Serre's parenthetic remark after Corollary 2 of Theorem 34 in 16.1 refers back to the earlier and more elementary step in his Corollary 2 of Proposition 42 in 14.4.
There are two essential steps here, based always on the fact that you have a triple $(K,A,k)$ involving a residue field $k$ of characteristic $p>0$ coming from a suitable ring $A$ whose field of fractions $K$ has characteristic 0. (The deeper results require some hypotheses on completeness, splitting fields, etc.) First you compare the Grothendieck groups of projective modules for $KG$ and $AG$ (as in Serre's Section 14). Then you build the $cde$-triangle as in Section 16 and obtain there the injectivity of the map $c$ as Geoff indicates. (The later book Methods of Representation theory I by Curtis-Reiner has a more detailed version of all this theory.)
Let me mention that working with a finite group is essential here, since a partial parallel exists for certain finite group schemes (in the guise of restricted Lie algebras) for which the matrix of Cartan invariants has determinant 0 and the behavior of projectives is more complicated.