Do you understand why the area of a circle is pi*r^2. Basically, cut it up into thin wedges where the circumference is 2*pi*r, then it can be re-arranged into length of thin wedges (triangles) of height r and area r/2*baselength. The sum of them must be (r/2)*2*pi*r as their bases correspond to the circumference
https://www.youtube.com/watch?v=YokKp3pwVFcIn the same way, the surface area of a sphere is 4*pi*r^2. You can "prove" this by equating the sphere to an open ended cylinder. Then the sphere is cut up into cones, each of volume r*basearea/3, just like the circle is cut up into wedges. The surface area of the sphere must equate to all the base areas of the cones, so the overall volume of the cylinder is
r*4*pi*r^2/3
or
4/3*pi*r^3
I'll have a quick dig round to see if I can find a pic of this.
Edit ...
https://www.researchgate.net/figure/Spherical-coordinates-r-th-and-ph-The-3-boldtype-coordinate-axes-represent-the-Cartesian_fig27_314913282Isn't perfect, but its ok. Imagine for each of the grey patches on the surface of the sphere, the cone which is generated by connecting the edges up to the center. This splits the sphere up into cones of height r where the surface area of the sphere is equal to the bases areas of the cones. So volume is
r/3*4*pi*r^2 as mentioned above
Much like the circle is cut up into pizza slices, I imagine the sphere being unrolled into a spiky sheet, where each spike is of height ~r and is one of the cones and the (base) sheet is area 4*pi*r^2.