This has really bothered me for a while. I understand fully the big picture idea of what is happening in Segal's "Categories and Cohomology Theories". You take a Segal space $X$ and you transform into a Spectra using his classifying space functor $B$ inductively to obtain $B^nX$ for all $n\geq 0$. This has an adjoint and it induces an adjunction on homotopy categories. An important part of the paper is that $X(1_+)$ is a homotopy associative, homotopy commutative $H$-space and this is easily deduced by using the the structure maps of the Segal category $\Gamma$.
There is one point I have failed to understand. The first involves the realization that Segal defines in his Appendix for simplicial spaces. How does this construction he uses work? The construction he gives is rather unclear and I don't understand why he used it or why it works. I am seeking some clarity on his choice of realization functor.
So, although Segal mentions this in passing, let me stress that there is a simpler inductive description of $||X||$ for a given bisimplicial set $X$. Namely, we put $||X||_{(-1)}=\varnothing$ and inductively define $||X||_{(n)}$ by a pushout square $$ \require{AMScd} \begin{CD} X_n\times\partial\Delta^n @>>> X_n\times\Delta^n\\ @V{f_n}VV @VVV\\ ||X||_{(n-1)} @>>> ||X||_{(n)} \end{CD} $$ Here, $f_n\colon X_n\times\partial\Delta^n\to ||X||_{(n-1)}$ is the map which on the $i$-th face of $\partial\Delta^n$ is defined by $X_n\times\Delta^{n-1}\xrightarrow{d_i\times\mathrm{id}}X_{n-1}\times\Delta^{n-1}\to||X||_{(n-1)}$. Finally, we put $||X||=\mathrm{colim}_n\,||X||_{(n)}$. Note the formal analogy with the definition of the skeleton of a simplicial set. I know this construction under the name of fat geometric realization of $X$. It is ''fat'' because if you follow Segal's first definition, you have yet to collapse some degenerate data in order to get the usual geometric realization. The fact that you don't collapse this structure means you have a ''fattened up'' version of the geometric realization.
This construction has the advantage that it is (depending on your familiarity with abstract homotopy theory) sort of immediately clear that the construction $||-||$ preserves weak homotopy equivalences, and hence is homotopically meaningful. Moreover, if we write $\mathrm{diag}(X)$ for the diagonal of $X$, i.e. the simplicial set $[n]\mapsto X_{n,n}$, there is a natural map $||X||\to\mathrm{diag}(X)$, which is a weak homotopy equivalence. (Via standard homotopy colimit arguments, you can reduce this to the case where $X=\Delta^{n,m}$ is a standard bisimplex, and then it is a matter of showing both sides are contractible.) But since $||-||$ preserves weak homotopy equivalences, this means that $\mathrm{diag}(-)$ does too (this is the Diagonal Lemma), and this was not at all obvious! It is a standard theme in homotopy theory that in order to study the homotopical properties of some construction, you look for a weakly equivalent construction which is easier to analyze, and in this case, $||-||$ is inductively defined via homotopy colimits, which means it is very easy to analyze some general properties of it.
There is a slightly different (but equivalent) approach to defining $BX$ for a $\Gamma$-space $X$, and then you can use the Diagonal Lemma to prove part of Segal's main result Proposition 1.4. In Segal's own paper, the usefulness of the fat geometric realization is for instance apparent in the proof of Proposition 1.6, where he uses the easy description of $||-||$ in terms of homotopy colimits to prove something about the usual geometric realization $|-|$. So, to summarize, the main point to using $||-||$ over any weakly equivalent construction is that its inductive definition in terms of homotopy colimits makes some standard arguments in abstract homotopy theory work cleanly, where they might not work at all if you use a different model of the same homotopy type.