I'll assume that you know that open balls in a metric space are open sets. Then since $x_1$ and $x_2$ are distinct$$r = \frac{1}{2} d(x_1,x_2) > 0.$$
Then the balls $B_{r}(x_1)$ and $B_{r}(x_2)$ are disjoint open sets containing $x_1$ and $x_2$ respectively. If there were a point $p$ in both balls then$$d(x_1,x_2) \leq d(x_1,p) + d(p,x_2) < r + r = d(x_1, x_2).$$Clearly this is impossible, so the sets are disjoint.
