Why do animals have more heavy nitrogen and carbon than plants?
Organisms accumulate certain isotopes because their metabolic enzymes have a slight preference for molecules containing that isotope. A well-known example is ribulose-1,5-bisphosphate carboxylase (Rubisco) which is responsible for CO$_2$ fixation in photosynthetic plants: this enzyme is somewhat more efficient with $^{12}$CO$_2$ than with $^{13}$CO$_2$, causing plants to accumulate $^{12}$C.

In animals, there is an enrichment for $^{15}$N because the deaminases and transaminases that transfer nitrogen from amino acids to urea (for excretion) tend to favor $^{14}$N. So in this case, $^{14}$N tends to be removed from the body faster than $^{15}$N when breaking down dietary protein. The differences are very small though: the relative enrichment of about 0.4% per trophic level corresponds to 0.367% $^{15}$N in a herbivore, compared to 0.366% $^{15}$N in a plant.

Source: this article.