"Mathematics" is NOT "science"! It makes no sense to ask "what is the science" of a given equation. Of course many things in science (as well as other fields) can be represented in mathematics so as equations but exactly what that equation is depends upon how you are "modeling" the science.
If you are asking "Why is that the equation of a circle", then you have to specify that you are representing x and y in a "Cartesian coordinate system" where the x and y axes are at right angles. Given a point (x, y), the line segment from (0, 0) to (x, 0) has length x and the line segment from (x, 0) to (x, y) has length y. By the "Pythagorean theorem" (still "mathematics", not "science"!) the distance from (0, 0) to (x, y) has length \(\displaystyle \sqrt{x^2+ y^2}\).
All the points on a circle with center at (0, 0) and radius r have distanced r from (0,0) (pretty much the definition of "circle") so \(\displaystyle r= \sqrt{x^2+ y^2}\) and. squaring both sides, \(\displaystyle r^2= x^2+ y^2\).