I'm not a fan of modulo math at this stage, but maybe what appears below might be helpful for the OP's request for "some 'informal' or more conceptually based justification of this theorem" (or maybe lead to something better from others).
Leaving aside math formatting technicalities, my current understanding of Wilson's Theorem (WT) is: p is prime iff [(p - 1)! + 1]/p = m where m is a positive integer.
It might be helpful to go back to Euclid's proof of infinitely many primes (as seen in https://en.wikipedia.org/wiki/Euclid%27s_theorem):
"Several variations on Euclid's proof exist, including the following:
The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them. Hence, n! + 1 is not divisible by any of the integers from 2 to n, inclusive (it gives a remainder of 1 when divided by each). Hence n! + 1 is either prime or divisible by a prime larger than n. In either case, for every positive integer n, there is at least one prime bigger than n. The conclusion is that the number of primes is infinite."
Whereas this proof indicates there is at least one prime > n up to (n! + 1), WT establishes that where a prime p happens to follow n, p will always have the 'magical' property of being evenly divisible into (n! + 1). I suppose that might very well be the part that the OP wanted elucidated in a less mathematical manner; but, unfortunately, I'm actually finding it difficult to translate what I'm seeing on paper etc into a proper descriptive form. I can "see" that the OP's query about (m - 1)! will be interesting to try examining without the modulo stuff, but that will have to wait.
The only other potentially helpful insight that currently comes to mind is the primes seeming to be the perfect example of the expression "A place for everything, and everything in its place". It's as if the whole number system was complete at the outset, such that the way things move forward seems to be partly based on what will happen later rather than being entirely based on what has happened in the past.
A big mistake in my prior answer that Jack M addressed was erroneously thinking (n! + 1) might yield only a single new prime that divides evenly. I subsequently tried something like 15! and confirmed three. I'll still avoid modulo math if possible, but at least I can come back here and read up on it if that's where my work leads.
With respect to the OP’s (m - 1)! query (even though he or she probably won't be back), here’s a hands-on approach that might be useful. Grab some paper and start writing the numbers. 2 might be thinking “What good is 1 anyway?”. 3 hits it off with 2 and they produce 6. Then 4 comes along to make 24, and we’ll skip over its idiosyncrasy. 5 sees a few strangers but likes to play, so we’re up to 120. Since you can see a prior 6, the next 6 presumably takes comfort in knowing someone’s cleared the way and happily ups things to 720. I'm not sure if 7 feels left out or superior, but it doesn't hold a grudge and brings the proceedings to 5,040. 8 already sees a buddy in 2x4, and our happy bunch is up to 40,320. 9 might find it odd not seeing both of its most familiar progenitors, but 3x6 are waiting to welcome it. ...
Building on the “intuition” aspect of the heading of the OP’s question, I haven’t regarded modulo math to be an important enough tool to put in my limited toolbox, but I have an appreciation of remainders. It occurred to me that in order for (p - 1)! to have a ‘magical’ property, there must be some kind of “overflow” from (p - 2)! . For the unfamiliar but interested, I would encourage you to give some thought to this. For those who want to dive into a pithy explanation of the infrequently mentioned extension of WT, see http://www.numericana.com/wilson.htm