In preparing a workshop on ordination techniques, I realized I was having the same difficulty in interpreting the eigenvalues of principal coordinate analysis for the same reasons that have puzzled you (@Paul Igor Costea), so I started digging around for some answers.
I have a few books on multivariate statistics that are not for the statistically faint of heart, and occasionally certain explanations get lost in some heavy matrix algebra (not the best for an 101 on ordinations). The best answer I found was actually in an overview of ordination methods for non-experts by Lengendre & Birks 2012 in a chapter of "Tracking Environmental Change using Lake Sediments".
The eigenvectors are typically much easier to interpret as they are essentially the coordinates (in reduced space) of a given object along a given axis. The eigenvalues, however, represent:
"the variance (not divided by degrees of freedom) of the objects along that axis."
(Lengendre & Birks 2012)
This is the most concise and precise interpretation I have found. While it is true that PCoA is not computed on a covariance matrix but on a distance matrix, PCoA and PCA are very similar, and the following simple example (from the same chapter) explains the mathematical relationship between the eigenvalues computed by each technique:
"From an object-by-variable data matrix
Y, compute matrix
D of Euclidean distances among the objects. Run PCA using matrix
Y and PCoA using matrix
D. The eigenvalues of the PCoA of matrix
D are proportional to the PCA eigenvalues computed for matrix
Y (they differ by the factor
(n – 1) [i.e.the degrees of freedom]), while the eigenvectors of the PCoA of
D are identical to matrix
F [i.e. the matrix of eigenvectors] of the PCA of
Y. Normally, one would not compute PCoA on a matrix of Euclidean distances since PCA is a faster method to obtain an ordination of the objects in
Y that preserves the Euclidean distance among the objects.This was presented here simply as a way of understanding the relationship between PCA and PCoA in the Euclidean distance case. The real interest of PCoA is to obtain an ordination of the objects from some other form of distance matrix more appropriate to the data at hand — for example, a Steinhaus/Odum/Bray-Curtis distance matrix in the case of assemblage composition data."