Consider lattice paths from $\langle 0,0\rangle$ to $\langle n-1,n\rangle$ in the planar lattice $\mathbb{Z}^2$ using unit steps to the right (i.e., $\langle i,j\rangle \to \langle i+1,j\rangle$) and unit steps to the top (i.e., $\langle i,j\rangle \to \langle i,j+1\rangle$).
For each path $\pi$ from $\langle 0,0\rangle$ to $\langle n,n-1\rangle$ let $r(\pi)$ be the smallest $j$ such that $\langle n,j\rangle$ is in $\pi$. For $j=0,\dots,n-1$ let $\Pi_j=\{\pi:r(\pi)=j\}$. Then $|\Pi_j|=\binom{n-1+j}j$ (since the elements in $\Pi_j$ are in bijection with lattice paths from $\langle 0,0\rangle$ to $\langle n-1,j\rangle$, of which there clearly are $\binom{n-1+j}j$), so there are
$$\sum_{j=0}^{n-1}|\Pi_j|=\sum_{j=0}^{n-1}\binom{n-1+j}j$$ paths from $\langle 0,0\rangle$ to $\langle n,n-1\rangle$.
Let $\Pi_j'$ be the paths from $\langle 0,0\rangle$ to $\langle n-1,n\rangle$ that pass through $\langle j,n\rangle$ but not $\langle j-1,n\rangle$; reflection in the diagonal gives a bijection between $\Pi_j'$ and $\Pi_j$, so there are $$\sum_{j=0}^{n-1}|\Pi_j'|=\sum_{j=0}^{n-1}\binom{n-1+j}j$$ paths from $\langle 0,0\rangle$ to $\langle n-1,n\rangle$.
Each path from $\langle 0,0\rangle$ to $\langle n,n\rangle$ passes through exactly one of the points $\langle n,n-1\rangle$ and $\langle n-1,n\rangle$, so
$$\binom{2n}n=2\sum_{j=0}^{n-1}\binom{n-1+j}j\;.$$
