There is no good definition of what is an "amount of entanglement". We have some requirements, such as saying that a measure of entanglement must be convex and cannot increase under local operations, but beyond that it is really a matter of taste.
There is a nice interpretation of entanglement negativity, though, in the case that $\rho^{T_A}$ only has a single negative eigenvalue. Let it be $-\lambda$. Then by construction $\mathcal N(\rho) = \lambda$, and $d\mathcal N(\rho)$ almost coincides with the amount of white noise you must add to $\rho$ before it becomes separable.
This is another measure of entanglement, called random robustness, defined more precisely as $R(\rho)$ being the minimal $s \ge 0$ such that the state $$\rho' = \frac1{1+s}(\rho + s I/d)$$ is separable.
I'm saying almost because $\rho^{T_A} \ge 0$ in general does not imply that $\rho$ is separable. But in the cases when it does, $R(\rho)$ is the minimal $s$ such that $${\rho'}^{T_A} = \frac1{1+s}(\rho^{T_A} + s I/d) \ge 0,$$ which is precisely $d\lambda$.
More generally, I don't know any nice interpretation for entanglement negativity. that is, the minimal eigenvalue of $\rho^{T_A}$