Let $N(\alpha,\beta)$ be the number of times the equation
$$
\alpha\cdot x \oplus \beta \cdot y=0 \tag{0} \label{0}
$$
holds. Then the LAT matrix entry is
$$
L(\alpha\cdot x \oplus \beta \cdot y )=
N(\alpha\cdot x \oplus \beta \cdot y)-2^{n-1}
$$
and since the second term is even (can prove $n=1$ case directly) we need to show that $N(\alpha\cdot x \oplus \beta \cdot y)$ is even.
The basic idea is that the set of vectors over $\mathbb{F}_2^n$ of the form
$$a\cdot x=c \tag{1} \label{1} $$ has even cardinality
for any constant $c\in \mathbb{F}_2.$ Observe that this applied to $\alpha\cdot x$ and $\beta\cdot y$ means that $N(\alpha,\beta)$ is also even if we can prove $\eqref{1}$.
To prove $\eqref{1}$ assume $a\neq 0$ in $\mathbb{F}_2^n,$ since if $a=0,$ the relation holds for all $x\in \mathbb{F}_2^n,$ i.e., for $2^n$ vectors.
If $a\neq 0,$ let $w$ be the Hamming weight of $a$ and let $s_a$ be the support of $a,$ i.e., $s_a=\{i: 1\leq i\leq n, a_i=1\}$. Then, the inner product $a\cdot x$ is invariant when the components $a_i$ in the complement of the support take on all possible $2^n-2^w$ values, since they don't feature in the inner product. Clearly $2^n-2^w$ is even.
Finally note that equation $\eqref{0}$ holds if and only if $\alpha\cdot x$ and $\beta\cdot y$ are both equal to the same constant $c$. So it holds an even number of times.
Remark: This argument also works for an arbitrary $n\times m$ S-box $S:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^m$ for any $m\geq 1$.
