Face Recognition: Simplistic Explanation on PCA Eigenface Algorithm
The step-to-step explanation in Eigenface seems quite clear to me.

A covariance matrix is like an high-dimensional extension of the variance, which is computed by removing the average from your only sample.


Yes, you remove the average face ($ \operatorname{AF}$ from all images, but keep it preciously. Your cov(image1) definition seems weird to me, but suppose you have it.
The eigenvectors of the big matrix are eigenfaces ($ \operatorname{EF}$), put in vector form, you can reshape them to image form. Imagine that as a common trait shared by several persons in your image base.


At this stage, each image $I_k$ can be approximated by a linear combination of $ \operatorname{EF}$s:
$$I_k =  \operatorname{AF}+a_k \operatorname{EF}_1+b_k \operatorname{EF}_2+c_k \operatorname{EF}_3\ldots\,.$$
The idea is that the different pictures of the same person will have almost the same $a_k,b_k,c_k$, because their share the same traits. But a different person would have quite different coefficients. So comparing the vectors of coefficients may help distinguish different persons. In your case:


You have computed eigenfaces. A new picture arrives. If you compute its coefficients on the $\operatorname{EF}$s (called a projection), if they are close to the coefficients of another person, in fact the closest, then, maybe, the new picture is a picture of the guy.


Additional links:


Eigenfaces, for Facial Recognition
Eigenfaces
Face recognition 101: Eigenfaces