It's straightforward to show that

$$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$

but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first $n$ natural numbers count the number of ways I can choose a pair out of $n+1$ objects? What's the intuition behind this?