This is explained very nicely in [Law77]. Here is a sketch.
We will use the following crucial lemma.
Lemma (Haefliger). Let $M$ be a smooth manifold and let $\mathcal{H}$ be a $\Gamma_q$-structure on $M$. There are a smooth fiber bundle $p:E\to M$ with fiber $\mathbb{R}^q$, a $\Gamma_q$-foliation $\mathcal{F}$ on $E$, and a continuous section $s:M\to E$ such that $s^*\mathcal{F}=\mathcal{H}$.
(Some people seem to call $E$ the foliated microbundle.)
Accepting the leemma for now, we can deduce assertion ($\ast$) from Bott's theorem, as follows. Let $c\in H^n(B\Gamma_q;\mathbb{R})$ be any element which is a polynomial of Pontryagin classes, where $n>2q$. We wish to show that $c=0$. Since every finite CW complex has the homotopy type of a smooth manifold, it suffices to show that for each smooth manifold $M$ and each map $f:M\to B\Gamma_q$, the pullback $f^*c$ is zero. Using the lemma (and replacing $M$ if necessary), we may assume further that $f$ classifies a $\Gamma_q$-foliation on $M$. But then Bott's theorem shows that $f^*c=0$, and the proof is complete $\square$
And here's a sketch proof of the lemma, for the sake of completeness.
(Proof of the lemma)
Represent $\mathcal{H}$ by a cocycle $(\{U_i\}_{i\in I}, \{\gamma_{ij}\}_{i,j\in I}),$ where $\{U_i\}_{i\in I}$ is a locally finite cover of $M.$ For each point $x\in M$, let $I(x)\subset I$ denote the set of indices $i\in I$ such that $x\in U_i.$ By the definition of $\Gamma_q$-structures, there are a neighborhood $V_x\subset \bigcap_{i\in I(x)}U_i,$ an open set $O_i^x\subset \mathbb{R}^q$ which contains $\gamma_i(x)$ and is diffeomorphic to $\mathbb{R}^q$ for each $i\in I(x)$, and diffeomorphisms $\phi_ij^x:O_i^x\to O_j^x$ such that $\gamma_ij(x')=[\phi_{ij}^x,\gamma_i(x')]$ for all $x'\in V_x$. ($[-,-]$ denotes a germ.) By shrinking the $O_i^x$'s and $V_x$ if necessary, we may assume that $\phi_{ii}^x=\operatorname{id}$ and $\phi_{jk}^x\phi_{ij}^x=\phi_{ik}^x$ for all $i,j,k\in I(x)$.
Now construct a smooth manifold $E$ from the disjoint union $\coprod_{i\in I, x\in U_i}V_x \times O_i ^x$ by identifying $(x,y)\in V_{x_0}\times O_{i_0}^{x_0}$ with $(x,\phi _{ii'}^x(y))$ if $x\in V_{x_0}\cap V_{x_1}.$ This $E$ and the evident projections, sections, and foliations of $E$, have the desired properties. $\square$
Reference
[Law77] Lawson, H. Blaine, Jr. The quantitative theory of foliations. Expository lectures from the CBMS Regional Conference held at Washington University, St. Louis, Mo., January 6–10, 1975 Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 27 American Mathematical Society, Providence, RI, 1977. v+65 pp.