Simple, intuitive high school level explanation of spectral theorem and the covariance matrix

Question

Ive been trying to understand spectral theorem ( a symmetric matrix has n independent, orthogonal eigenvectors ), but the proof seems to be way above my level. Similarly, ive been trying to realize the geometric significance of the covariance matrix and why its eigenvectors are the directions of maximum variance, and the eigenvalues are the magnitude of variance. Any help would be appreciated. Thank you.