(1/2)×θ×r^2

gives you the area of a sector circle, like a slice of pie.

(1/2)×θ×side1×side2

is the side-angle-side formula for the area of a triangle.

We know that the triangle has two sides that are equal to the radius, so we replace side1×side2 with r^2. Since the area of the arc segment is equal to the area of a sector minus the triangle encompassed by the sector, we can subtract triangle area from sector area to get

(1/2)×(θ-sin(θ))×r^2