What a kernel remembers

A homomorphism's kernel records exactly which elements become indistinguishable from the identity.

Two elements $g,h\in G$ have the same image precisely when $h^{-1}g$ lies in the kernel. Thus the fibers of a group homomorphism are its kernel's cosets.

The First Isomorphism Theorem says: once these indistinguishable elements are identified using Quotient Groups, what remains is exactly the image.