Assume that $p\equiv \pm 2 \pmod{5}$ and consider the following calculation:
$$\left( \frac{p}{5} \right) = \color{red}{\left( \frac{\pm 2}{5} \right) \equiv (\pm 2)^\frac{5-1}{2}} \equiv 4\equiv 1\pmod{5}$$
What is the explanation for the red-colored equivalence? (can it be generalized or is it some special case?)
It's the Euler's Criterion, which is one of the most important tools in quadratic reciprocity. You can find a proof of it in the Wikipedia's page.