The combinatorial definition of the Schur functions is
$$
s_\lambda(x) = \sum_{T \in SSYT(\lambda)} x^{cont(T)}
$$
where $SSYT(\lambda)$ is the set of semi-standard Young tableaux of shape $\lambda$ and $x^{cont(T)}$ is the product over all $i$ of  $x_i^{\# i\text{'s in }T}$. This is not manifestly a symmetric function. The Bender-Knuth involution proves that $s_\lambda(x)$ is invariant after swapping $x_i$ with $x_{i+1}$, and thus $s_\lambda(x)$ is, indeed, symmetric.