Let me try a similar explanation with different words. (Note that my explanation does not cover characteristic $2$.)
A quadratic form is nondegenerate if any of its associated symmetric matrices has nonzero determinant. (Alternately, if the associated bilinear form $B(x,y) = q(x+y) - q(x) - q(y)$ is nondegenerate in the usual sense: $B(x,y) = 0 forall y in K implies x = 0$.)
Let $K$ be a field of characteristic different from $2$. The hyperbolic plane is the special quadratic form
H(x,y) = xy.
(As with any quadratic form over $K$, it can be diagonalized: $frac{1}{2} x^2 - frac{1}{2} y^2$.)
A nondegenerate quadratic form $q(x_1,ldots,x_n)$ is isotropic if there exist $a_1,ldots,a_n in K$, not all $0$, such that $q(a_1,ldots,a_n) = 0$ and otherwise anisotropic.
Witt Decomposition Theorem: Any quadratic form $q$ can be written as an orthogonal direct sum of an identically zero quadratic form, an anistropic quadratic form, and some number of hyperbolic planes. In particular, any isotropic quadratic form $q(x_1,...,x_n)$ can be written, after a linear change of variables, as $x_1 x_2 + q(x_3,...,x_n)$.
For your purposes, you might as well assume your quadratic form is nondegenerate -- otherwise, it simply involves more variables than actually appear!
Now over a finite field, the Chevalley-Warning theorem implies that any nondegenerate quadratic form in at least three variables is isotropic, so that by Witt Decomposition, you can split off a hyperbolic plane. If you still have at least three variables, you can do this again. Repeated application gives your result.
References:
For Chevalley-Warning:
http://math.uga.edu/~pete/4400ChevalleyWarning.pdf
For Witt Decomposition:
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