This expression of p and q is called Cantor pairing function. It maps pair of positive integers onto positive integers in a one-to-one manner. It is useful because it demonstrates that there are as many positive integers as pairs of positive integers.
To understand why it works, consider this picture. The value of the pairing function is shown near each point. Arrow show the direction in which the value of the function increases.
Note that p + q is constant on the diagonals. Let's look at the blue diagonal, where p + q = 5. How many points come before, i.e., how many points are in the green triangle? It's 1 + 2 + 3, or, in terms of p and q, it's 1 + 2 + ... + (p + q) - 2 = 1/2 [(p + q - 2) * (p + q - 1)] (as the sum of an arithmetic progression). If we add the vertical coordinate q, we get the value of the function on the blue diagonal.