Think of your $n+1$ items as being labeled from 1 to $n+1$. You need to make two choices if you're picking two elements: The one with the smaller label, and the one with the larger label. If you pick $n$ for the smaller label, there's only 1 choice for the larger. If you pick $n-1$ for the smaller, there's 2 choices for the larger, etc., all the way down to $n$ choices for the larger if you choose $1$ for the smaller. The total number of ways you can pick two elements is then the sum of the ways you can do it for each choice of smaller label, which is $1 + 2 + ... + n$.