What does a symmetric matrix transformation do, geometrically?

A real symmetric matrix is always orthogonally diagonalizable, meaning that there's a basis for $\mathbb R^n$ consisting of mutually perpendicular eigenvectors of the matrix. Thus you can understand multiplying a column vector by a symmetric matrix geometrically as:

• Express the input vector in a different rectangular coordinate system that depends on the matrix.
• Multiply each coordinate by some constant that depends on which axis in the new coordinate system it corresponds to -- that is, stretch, shrink or flip each axis independently of each other.
• Express the result back in the original coordinate system.