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:

http://math.uga.edu/~pete/quadraticforms.pdf