Note: This problem is intuitively the same as the handshake problem where instead of counting the max handshakes among $n$ people you want to find the max edges among $n$ vertices.
Suppose we have $n$ vertices labeled $v_i$, $(i = 1,...,n)$ (the actual order is irrelevant) :
Step 1) Take vertex $v_1$ and connect it to all other $(n-1)$ vertices, now disgard $v_1$.
Step 2) Take vertex $v_2$ and connect it to all other $(n-2)$ vertices, now disgard $v_2$.
$$...$$
Step n-2) Take $v_{n-2}$ and connect it to all other $(2)$ vertices, now disgard $v_{n-2}$.
Step n-1) Take $v_{n-1}$ and connect it to all other $(1)$ vertices, now disgard $v_{n-1}$.
Now, notice the number of edges (in parenthesis) above we add in each step can be written as:$$\sum\limits_{i=1}^{n-1} i = \frac{n(n-1)}{2}$$