I believe your question is: What is the importance of being nuclear of order zero for the transfer (=Perron-Frobenius) operator in Ergodic theory?

The importance of being nuclear is a theorem by Grothendieck: If $L$ is nuclear of order less or equal than $2/3,$ then $det(I-zL)$ is entire in $z$ and we have that $det(I-zL)=\prod_n(1-z\lambda_n)$ where $\lambda_n$ are the eigenvalues of $L.$ From this and the trace $tr(L)=\sum_n\lambda_n$ we can write $\log det(I-zL)=-\sum_n \frac{z^n}{n}Tr(L^n).$ This is quite helpfull because of many reasons. The main reason is that Ruelle succeeded to prove the nuclearity (of order zero) of the Perron-Frobenius operator associated to Axiom A diffeomorphisms in the seventies, so by Grothendieck's theorem he related the eigenvalues of the Perron-Frobenius operator with the inverses of the zeros of $\exp(-\sum_n \frac{z^n}{n}Tr(L^n)).$ This motivates many accurate approximation results in Ergodic theory by Pollicott, as eigenvalues of the Perron-Frobenius operator are known to be related to mixing properties of the associated dynamical system, and on the other hand, the inverses of the zeros of $\exp(-\sum_n \frac{z^n}{n}Tr(L^n))$ can be approximated using the Taylor expansion of holomorphic functions. A few examples of these mixing properties are the following:

- Invariant densities are fixed points of the Perron-Frobenius operator.
- If there exist an ergodic invariant density, then the first eigenvalue of the Perron-Frobenius operator is equal to 1 and it is simple.
- If there exist a mixing invariant density, then the first eigenvalue of the Perron-Frobenius operator is equal to 1 and moreover, every other eigenvalue has modulus strictly less than 1.

I suggest you to read Pollicott, Baladi and Ruelle, in this order. They have some lecture notes available on the web. On the other hand, I do not think you will find anything useful directly from Grothendieck's 1956 notes, Ruelle did it!

Another reason of its importance (of $L$ being nuclear) it is that isolated eigenvalues are stable under perturbations (in Kato's sense). Therefore, as a consequence, one can usually use that $L$ is nuclear in order to prove stability results. For example, of the invariant density.