Proof

Consider the question: How many lattice paths are there from $(0,0)$ to $(n,n)\text{?}$

Answer 1: We must travel $2n$ steps, and $n$ of them must be in the up direction. Thus there are ${2n \choose n}$ paths.

Answer 2: Note that any path from $(0,0)$ to $(n,n)$ must cross the line $x + y = n\text{.}$ That is, any path must pass through exactly one of the points: $(0,n)\text{,}$ $(1,n-1)\text{,}$ $(2,n-2)\text{,}$ …, $(n, 0)\text{.}$ For example, this is what happens in the case $n = 4\text{:}$

$A light gray grid with points (0,0) and (4,4) identified in the lower left and upper right corners respectively. The line x + y = 4 is shown extending from the top left to bottom right corner. The line intersects the labeled points (0,4), (1,3), (2,2), (3,1), and (4,0).$

How many paths pass through $(0,n)\text{?}$ To get to that point, you must travel $n$ units, and $0$ of them are to the right, so there are ${n \choose 0}$ ways to get to $(0,n)\text{.}$ From $(0,n)$ to $(n,n)$ takes $n$ steps, and $0$ of them are up. So there are ${n \choose 0}$ ways to get from $(0,n)$ to $(n,n)\text{.}$ Therefore there are ${n \choose 0}{n \choose 0}$ paths from $(0,0)$ to $(n,n)$ through the point $(0,n)\text{.}$

What about through $(1,n-1)\text{.}$ There are ${n \choose 1}$ paths to get there ($n$ steps, 1 to the right) and ${n \choose 1}$ paths to complete the journey to $(n,n)$ ($n$ steps, $1$ up). So there are ${n \choose 1}{n \choose 1}$ paths from $(0,0)$ to $(n,n)$ through $(1,n-1)\text{.}$

In general, to get to $(n,n)$ through the point $(k,n-k)$ we have ${n \choose k}$ paths to the midpoint and then ${n \choose k}$ paths from the midpoint to $(n,n)\text{.}$ So there are ${n \choose k}{n \choose k}$ paths from $(0,0)$ to $(n,n)$ through $(k, n-k)\text{.}$

All together then the total paths from $(0,0)$ to $(n,n)$ passing through exactly one of these midpoints is

\begin{equation*} {n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + \cdots + {n \choose n}^2\text{.} \end{equation*}

Since these two answers are answers to the same question, they must be equal, and thus

\begin{equation*} {n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + \cdots + {n \choose n}^2 = {2n \choose n}\text{.} \end{equation*}