Step 1: The key fibration. Consider the Adams operation $\psi^q: BU \to BU$ at the level of the infinite unitary group. Define $F\psi^q$ as the homotopy fiber of $\psi^q - 1$:

$F\psi^q \xrightarrow{} BU \xrightarrow{\psi^q - 1} BU.$

This gives a fibration sequence. The homotopy groups of $BU$ are:

$\pi_n(BU) = \begin{cases} \mathbb{Z} & n \text{ even} \\ 0 & n \text{ odd} \end{cases}$

The map $\psi^q - 1: \pi_{2i}(BU) \to \pi_{2i}(BU)$ is multiplication by $q^i - 1$ on $\mathbb{Z}$. From the long exact sequence of the fibration:

$\cdots \to \pi_{2i}(BU) \xrightarrow{q^i - 1} \pi_{2i}(BU) \to \pi_{2i-1}(F\psi^q) \to \pi_{2i-1}(BU) = 0$

we get $\pi_{2i-1}(F\psi^q) \cong \mathbb{Z}/(q^i - 1)\mathbb{Z}$ and $\pi_{2i}(F\psi^q) = 0$ for $i \geq 1$, with $\pi_0(F\psi^q) = 0$.

Step 2: The Brauer lift factors through $F\psi^q$. Quillen shows that the Brauer lift $\phi: BGL(\mathbb{F}_q) \to BU$ satisfies $(\psi^q - 1) \circ \phi \simeq *$ (null-homotopic). This is because the Frobenius $\operatorname{Frob}_q$ acts on the representation ring of $GL_n(\overline{\mathbb{F}}_q)$ by $\psi^q$, and representations over $\mathbb{F}_q$ are exactly the Frobenius-fixed points, giving $\psi^q(\tilde{\rho}) = \tilde{\rho}$ for representations coming from $\mathbb{F}_q$.

Thus $\phi$ lifts to a map

$\tilde{\phi}: BGL(\mathbb{F}_q) \to F\psi^q.$

Step 3: The map $\tilde{\phi}^+$ is a homotopy equivalence. Applying the plus construction to the source (with respect to $E(\mathbb{F}_q)$), we get

$\tilde{\phi}^+: BGL(\mathbb{F}_q)^+ \to F\psi^q.$

To show this is an equivalence, Quillen proves it induces an isomorphism on homology with all finite coefficients $\mathbb{Z}/m\mathbb{Z}$.

Step 3a: Cohomology of $BGL_n(\mathbb{F}_q)$. Using the theory of finite groups of Lie type, Quillen computes

$H^*(BGL(\mathbb{F}_q); \mathbb{F}_\ell) \cong H^*(BU; \mathbb{F}_\ell)^{\psi^q = 1}$

for primes $\ell \neq p$. This is the $\psi^q$-fixed part of the polynomial ring $\mathbb{F}_\ell[c_1, c_2, \ldots]$.

Step 3b: Comparison. The cohomology of $F\psi^q$ with $\mathbb{F}_\ell$ coefficients can be computed from the Serre spectral sequence of the fibration. One obtains the same answer as in Step 3a, and $\tilde{\phi}^+$ induces the identity on these cohomology rings.

Step 3c: The $p$-primary part. For $\ell = p$ (the characteristic), one shows separately that $H^*(BGL(\mathbb{F}_q); \mathbb{F}_p) \cong H^*(F\psi^q; \mathbb{F}_p)$ via the Brauer lift.

Step 4: Conclusion. By the Whitehead theorem (for simple spaces), since $\tilde{\phi}^+$ induces isomorphisms on all homology groups with finite coefficients, and both spaces have finitely generated homotopy groups, $\tilde{\phi}^+$ is a homotopy equivalence:

$BGL(\mathbb{F}_q)^+ \simeq F\psi^q.$

Therefore $K_n(\mathbb{F}_q) = \pi_n(BGL(\mathbb{F}_q)^+) = \pi_n(F\psi^q)$, which gives the stated formula. $\square$