Probability mass function of binomial distribution is$$
{n \choose k} p^k (1-p) ^{n-k}
$$
probability density function of beta distribution is
$$
\frac{1}{\mathrm{B}(\alpha, \beta)} p^{\alpha-1} (1-p)^{\beta-1}
$$
we can rewrite
$$
{n \choose k} = \frac{1}{(n+1) \mathrm{B}(k+1, n-k+1)}
$$
if we substitute $k+1 = \alpha$ and $n-k+1 = \beta$ then pmf
of binomial distribution becomes
$$
\frac{1}{(n+1) \mathrm{B}(\alpha, \beta)} p^{\alpha-1} (1-p)^{\beta-1}
$$
so beta can be understood as a distribution of $k/n$ proportions in
$n$ trials where the average proportion is denoted as $\mu$
$$
\frac{1}{\mathrm{B}(n\mu, n(1-\mu))} p^{n\mu+1} (1-p)^{n(1-\mu)+1}
$$
Alternatively $n$ may be understood as precision parameter.