Every simple undirected graph with more than $(n-1)(n-2)/2$ edges is connected
I am not sure what bothers you but as I see it you are confused about the following two facts 


If a graph is connected then $e \geq n-1.$
If a graph has more than $e &gt; \frac{(n-1)(n-2)}{2}$ then it is connected.


Notice that the implications in 1 and 2 are in opposite directions.

For a proof of 2. you can check out this link.