Loadings to reproduce Correlation Coefficients between original variables? 
The textbook I have says that, in factor analysis, loadings can be used to reproduce the correlation coefficients between original variables, as it is depicted in the image.
For instance, the book says, among 6 variables, the correlation coefficient between v1 and v6 can be expressed as below (the $\gamma$16 part), which I totally cannot understand.
I really appreciate any help.

This is the implied correlation coefficient. If the model is not saturated, it will not be equal to the actual correlation coefficient (although you hope it will be close, and the better the model, the closer it will be).
You seem to have the answer - I'm not sure what to add (but I'll try).
$\gamma_{1,6}$ is the (implied correlation between v1 and v6. Using path tracing rules:
$\gamma_{1,6} = \lambda_{2, 1} \times \lambda_{6, 1} + \lambda_{1, 2} \times \lambda_{6, 2}$
If your loadings are all $0.6$, then it's:
$\gamma_{1,6} = 0.6 * 0.6 + 0.6 * 0.6 = 0.72$
