What is the Katz-Sarnak philosophy?

The "Katz-Sarnak philosophy" is just the idea that statistics of various kinds for $L$-functions should, in the large scale limit, match statistics for large random matrices from some particular classical compact group.

First you need to decide what kinds of zeros to look at: the high zeros of an individual $L$-function, the low zeros (near the real axis) in some family of related $L$-functions (e.g., low zeros in all Dirichlet $L$-functions, maybe with a restriction on the parity of the character), and so on. The choice of what sort of zeros you look at can have an effect, e.g., looking at statistics of the low zeros in a family can reveal a "symmetry" (similar statistics to a specific classical compact group) that is not evident when looking at high zeros of an individual $L$-function (universality, not distinguishing between different $L$-functions).

Next you need to normalize the (nontrivial) zeros. For example, zeros of an individual $L$-function tend to get close to each other high up the critical line, so you count the number of zeros up to height $T$ and then rescale them so they get average spacing 1. It's those rescaled zeros that you work with when computing your various statistics.

Next you need to work with a suitable class of test functions and be able to actually carry out statistical calculations to reveal similarity with some class of random matrices.

Part of the motivation for people to look at these questions is the hope that it could suggest a spectral interpretation of the nontrivial zeros. Look up the Hilbert-Polya conjecture (Hilbert had nothing to do with it).

You mentioned the Katz-Sarnak book in your question. Keep in mind that this book is entirely about $L$-functions for varieties over finite fields, not the ones like $\zeta(s)$ that are associated to number fields. The blog post and paper you mention by Kowalski and Miller are about $L$-functions over number fields, and that is perhaps why they look very different to you from the Katz-Sarnak book. Another paper where the number field case is treated head-on is Rudnick, Sarnak, "Zeros of principal $L$-functions and random matrix theory," Duke Math. J. 81 (1996), 269–322. If you want to see a textbook's treatment of the relation between $L$-functions over number fields (like $\zeta(s)$ and Dirichlet $L$-functions) and random matrix theory, consider the book "An Invitation to Modern Number Theory" by Steve Miller and Ramin Takloo-Bighash. The random matrix theory is in Part 5.

I'm going to give an answer that discusses some things that the other answers don't go into as much detail on. In particular let me try to explain why the results you mention on classical groups having the same $n$-level correlations are in fact relevant to the number theory situation. It will build off KConrad's answer in some ways.

The Katz-Sarnak book proves theorems about zeroes of $L$-functions in function field setting. These are defined by exact analogues of the definitions of $L$-functions, over another field. Of course, as KConrad points out, these aren't the same as $L$-functions over a number field, and many of the families considered in the Katz-Sarnak book don't match the families of interest over number fields. But some match, and this has been rectified in later work.

One key advantage of the function field setting is that the situation is conceptually clearer. The $L$-function is a polynomial, and by the Grothendieck-Lefschetz trace formula it is the characteristic polynomial of some naturally defined matrix (the action of a Frobenius element on a Tate module or etale cohomology group). So we have no difficulty interpreting the zeroes as eigenvalues of a matrix. Moreover, this matrix lies in some matrix group, the monodromy group of the family.

These matrices are over an $\ell$-adic field and so can't naturally be viewed as unitary matrices. But Deligne (with Weil first in some cases) showed that if we base change to the complex numbers, they are conjugate to unitary matrices. So in fact the zeroes of the $L$-function are eigenvalues of elements of the maximal compact subgroup of some algebraic matrix group. So in this setting, the question of why we should expect the zeroes to behave like eigenvalues of random unitary matrices is not mysterious at all - it's the simplest possible behavior they could have, given that they are eigenvalues of unitary matrices.

In fact Deligne did more and showed that these Frobenius conjugacy classes are equidistributed in the maximal compact subgroup of the monodromy group. In other words, the statistics of any continuous function of the set of zeroes at all matches the statistics for random matrices in some compact group. Such a powerful result doesn't come without a cost, and it has two. First, we may not know what the group is. Second, this statistical equivalence only shows up in the limit as we take the finite field size to infinity. In this limit, the number of zeroes is fixed, so we can take the same compact group each time. However, in the number field settings, people always consider limits where some parameter controlling the number of zeroes grows.

The first cost was dealt with by Katz-Sarnak by computing the group in some important cases. They found that, while a priori it could be any group at all, in fact it was usually a classical group (which is something Katz had already noticed in some of his earlier investigations). So they focused attention on the maximal compact subgroups of classical groups.

The second was and is more difficult to deal with. However, since the transfer from the function field case to the number field case is conjectural anyways, one merely has to guess how to transfer statistics from finitely many zeroes to infinitely many zeroes, and then perform the calculations to do the transfer. They guessed that taking the limit of distribution of the smallest $k$ eigenvalues in a family of matrices in larger and larger groups would match the distribution of the smallest $k$ zeroes in $L$-functions over number fields. The first few chapters of the Katz-Sarnak book are then devoted to calculating these limits, which is the part that looks like honest random matrix theory.

So Katz and Sarnak combined algebraic geometry - the results of Deligne and the monodromy group calculations - with random matrix theory - these calculations in the limit as the size of the classical group grows - to obtain precise predictions for the statistics of zeroes of $L$-functions over number fields. These predictions have then been expanded and refined in later work.

I guess this is all a bit removed from your motivating question, which is how that random matrix theory can help in number theory. Unfortunately I don't know how much cutting edge research in random matrix theory is really needed, because we know of no rigorous relationship between $L$-functions over number fields and any kind of random matrix, so thus far random matrix theory has mostly been used to give predictions, and the predictions we already have are hard enough to try to prove that any property of random matrices found by more powerful techniques might be even further out of reach over number fields. However, on a positive note, I would suggest you check out Terry Tao's latest blog post, which is on a question involving random matrices, still unsolved, motivated by number theory (in fact, possibly motivated by a joke I made in a talk?).

I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgomery, with some help from Dyson, that the Riemann zeros have the same correlation function as the eigenvalues of unitary random matrices. Some averages over the zeta function agree with averages over the unitary group. There was a major result in this area by Jon Keating and Nina Snaith, for example, when they were able to arrive at a formula related to the quantity $\int |\zeta(\frac{1}{2}+it)|^{2k}dt$ for positive integers $k$, which was previously unavailable.

(if you look at papers that cite JP Keating, NC Snaith, Random matrix theory and ζ (1/2+ it). Communications in Mathematical Physics 214, p. 57 (2000), you will find a lot of information about this).

The general idea is that the Riemann zeros have some universal statistical properties combined with some specific properties which are related to the prime numbers. On the RMT side there are no prime numbers, so only the universality aspect can be recovered. There is however another connection, which is to the quantum mechanics of chaotic systems. Energy levels of such systems have some universal statistical properties, described by RMT, and also some specific properties, related to the corresponding classical dynamics, so that periodic orbits are the analogue of prime numbers in this case. There is some nice work by Michael Berry and Jon Keating about this (such as M. Berry, J. Keating, The Riemann Zeros and Eigenvalue Asymptotics. SIAM Rev., 41(2), 236 (1999)) .

The relation between number theory averages and RMT averages extends in some ways (to other $L$-functions and their derivatives, to non-integer moments, etc.) I would venture that the KS philosophy is the idea that there exist universal statistical properties underlying some number theory questions, and that these are well described by RMT.