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 $$ \begin{align*} \tilde{a}_{00} &= \frac{a_{00} + a_{01} + a_{10} + a_{11}}{2}, \\ \tilde{a}_{01} &= \frac{a_{00} - a_{01} + a_{10} - a_{11}}{2}, \\ \tilde{a}_{10} &= \frac{a_{00} + a_{01} - a_{10} - a_{11}}{2}, \\ \tilde{a}_{11} &= \frac{a_{00} - a_{01} - a_{10} + a_{11}}{2}. \end{align*} $$
This is exactly the tensor square of the $2\times 2$ Hadamard matrix.