How is this definition of the quantum Fourier transform to be understood?
The index $\mathbf{k}$ goes over all vectors in $\{0,1\}^n$ (which can be identified with $\mathbb{Z}_2^n$). The index $\mathbf{x}$ also has the same range, and $\mathbf{k} \cdot \mathbf{x} = \sum_{i=1}^n k_i x_i$. You can think of this sum as being computed modulo 2 (or in $\mathbb{Z}_2$) if you wish, since $(-1)^2 = 1$.

As an example, when $n = 2$ we get 
$$
\begin{align*}
\tilde{a}_{00} &amp;= \frac{a_{00} + a_{01} + a_{10} + a_{11}}{2}, \\
\tilde{a}_{01} &amp;= \frac{a_{00} - a_{01} + a_{10} - a_{11}}{2}, \\
\tilde{a}_{10} &amp;= \frac{a_{00} + a_{01} - a_{10} - a_{11}}{2}, \\
\tilde{a}_{11} &amp;= \frac{a_{00} - a_{01} - a_{10} + a_{11}}{2}.
\end{align*}
$$

This is exactly the tensor square of the $2\times 2$ Hadamard matrix.