If $\mathbf{x} \sim N(\mathbf{0,I})$ and $\mathbf{y} = \mathbf{Ax}$, what does $\mathbf{A}^T \mathbf{A}$ represent? If $\mathbf{x} \sim N(\mathbf{0,I})$ then $\mathbf{AA}^T$ is the covariance matrix of $\mathbf{y} = \mathbf{Ax}$, but what does $\mathbf{A}^T \mathbf{A}$ represent?
In some places I have seen statements like "if $\mathbf{X}$ is the data, then $\mathbf{X}^T \mathbf{X}$ is (proportional/related to) the covariance matrix of the data." But here $\mathbf{A}$ is not data, it is a matrix of coefficients.
I was dealing with a problem where I had to reduce the size of $\mathbf{y}$ from $n$ to $1$ and it was solved via a PCA for $\mathbf{A}^T \mathbf{A}$ such that the reduce one-dimensional $y^*$ is given by $y^* = \mathbf{b}^T\mathbf{x}$ where $\mathbf{b}$ is the (normalised) eigenvector corresponding to the largest eigenvalue of $\mathbf{A}^T \mathbf{A}$. 
 A: Expansion of the term
Since, based on your comments, $\mathbf{A}$ has dimensions $n*m$ $\mathbf{A}^T \mathbf{A}$ expands to:
$$ A_{1,1}*A_{1,1} + …A_{n,1}*A_{n,1} ,... A_{1,1}*A_{1,m} + …A_{n,1}*A_{n,m}$$
$$ A_{1,m}*A_{1,1} + …A_{n,m}*A_{n,1} ,... A_{1,m}*A_{1,m} + …A_{n,m}*A_{n,m}$$
Interpretation of the expansion
The diagonal is the sum of squares for the coefficients of your regression, so the larger the magnitude of the contribution a coefficient is to your relationship, the larger the value will be. 
Limitations on the interpretation
Initially this sounds like a useful way to gauge the importance of each variable to the regression. What this actually means will depend heavily on your data types and pretreatment of variable prior to the regression and so is not usually used to assess variable importance. There is a lot of literature out there on variable importance and contribution.
https://www.sciencedirect.com/science/article/pii/S0951832015001672
https://aichamp.wordpress.com/2017/03/26/variable-importance-and-how-it-is-calculated/
http://blog.minitab.com/blog/adventures-in-statistics-2/how-to-identify-the-most-important-predictor-variables-in-regression-models
