Why do normals exist? I know what normals are, because I took physics.  However, I have this question: why do they exist in blender?

The only use I've ever had for them is negative side effects.  They make my mesh render funny.  Why doesn't blender simply treat both sides of the mesh the same?

What were the intentions of the developers including normals in the program, and not treating both sides of the mesh as if they had the normal?
A normal is a vector in 3D-space that is perpendicular to two other vectors (determinded by two edges of a face). It is calculated by the cross prduct.

A basic use of normals is in shaders e.g. to calculate reflections



Imgage: https://en.wikipedia.org/wiki/Reflection_(physics)

Normals could also be used for backface culling. This is a performance optimization which allows to skip rendering of faces which will not be visible (normals pointing away from to the camera). OpenGl uses a different method for that described here.

Left image includes backfaces, the right one doesn't.