I'm learning Principal Component Analysis (PCA) and came to know that eigenvectors of the covariance matrix of the data are the principal components, which maximizes the variance of the projected data. I understand the intuition behind why we need the variance of projected data as large as possible.

From this answer, I don't understand the following line:

The unit vector $u$ which maximizes variance $u^TΣu$ is nothing but the eigenvector with the largest eigenvalue.

I know how the variance of projected data points is $u^TΣu$ from this answer. But I don't understand why this will be maxed when $u$ is selected as eigenvectors of $u^TΣu$ with the highest eigenvalues?

Intuitively I see eigenvectors as the vectors which stay fixed in their direction under the given linear transformation (values may scale, which are known as eigenvalues). Source: This answer. and this video.

I can't relate why vectors with a fixed direction under given linear transformation give the highest variance? Any intuitive explanation will help! Thanks.