My favorite explanation for Moebius Inversion is the same as my favorite explanation of a lot of things: It's secretly linear algebra!
Call a poset $(P, \leq)$ "locally finite" if for any $a, b \in P$ the set $[a,b] = \{ x \mid a \leq x \leq b \}$ is finite.
Now for any locally finite $P$ let's look at the Incidence Algebra of $P$, defined as
$$
I(P)
\triangleq
\{ \alpha : P \times P \to \mathbb{C} \mid \alpha(x,y) = 0 \text{ if } x \not \leq y \}
$$
This might feel like it's being pulled out of a hat (and it is), but you asked for "knowledge garnered from more advanced theory", and the price we pay for that more advanced theory is that sometimes it's not a priori clear why we choose the definitions we do. But trust that this can be motivated, and let's carry on with the show.
Now, $I(P)$ is obviously a vector space over $\mathbb{C}$, and we can identify our choices of $\alpha$ with square matrices! The columns and rows are indexed by $P$, and the entry $A_{x,y} = \alpha(x,y)$. Notice now what matrix multiplication looks like:
$$(\alpha \star \beta)(x,y) = \sum_z \alpha(x,z) \beta(z,y)$$
but now the definition of $I(P)$ begins to motivate itself! Since $\alpha(x,z) = 0$ whenever $x \not \leq z$ and $\beta(z,y) = 0$ whenever $z \not \leq y$, the local finiteness of our poset ensures this sum is actually finite!
$$(\alpha \star \beta)(x,y) = \sum_{x \leq z \leq y} \alpha(x,z) \beta(z,y)$$
Of course, in the divisibility lattice, we get a special case
$$(\alpha \star \beta)(m,n) = \sum_{m \ \mid \ d \ \mid \ n} \alpha(m,d) \beta(d,n)$$
So now matrix multiplication is the secret to many of the convolution formulas you've seen! Before we move on, we'll want to know when we can invert matrices in $I(P)$, and a bit of work shows
$\alpha$ is invertible if and only if for all $x \in P$, $\alpha(x,x) \neq 0$.
Now, there's an obvious "canonical" element of $I(P)$ which we can associate with our poset. For historical reasons, we call it $\zeta$, and it's defined as follows:
$$
\zeta(x,y) = \begin{cases} 1 & x \leq y \\ 0 & \text{else} \end{cases}
$$
Notice that, $\zeta(x,x) = 1 \neq 0$ for all $x$. So $\zeta$ must be invertible in $I(P)$! Obviously you can see the punchline coming, but we write
$$
\mu \triangleq \zeta^{-1}.
$$
What good is this, though? Well, let's consider a sum
$$
f(n) = \sum_{d \mid n} g(d)
$$
We recognize this is a matrix-vector multiply. If $\vec{g} \triangleq (g(1), g(2), g(3), \ldots)^T$ and $\vec{f} \triangleq (f(1), f(2), f(3), \ldots)^T$, then we find
$$
\vec{f} = \zeta \vec{g}
$$
since, if we look at the $n$th coordinate of $\vec{f}$, we see
$$
f(n) = \sum_{1 \mid d \mid n} \zeta(1,d) g(d) = \sum_{d \mid n} 1 \cdot g(d)
$$
Of course, now moebius inversion becomes almost tautological!
If $\vec{f} = \zeta \vec{g}$, then we must have $\vec{g} = \zeta^{-1} \vec{f} = \mu \vec{f}$.
And indeed, we recover the moebius inversion formula by looking at the $n$th component of $\vec{g}$:
$$
g(n) = \sum_{1 \mid d \mid n} \mu(1,d) \cdot f(d)
$$
and with some massaging, this formula turns into the "classical" formulation of Moebius Inversion.
The theory of incidence algebras is extremely cool, and is a very powerful technique in algebraic combinatorics. As an exercise, you might be interested in checking for yourself that the theory of moebius inversion in the powerset lattice recovers the principle of inclusion/exclusion! You might start with a small powerset, and manually invert $\zeta$ (or ask a computer) to compute $\mu$, then check that PIE falls out.
This is all very effective, and there are explicit formulas which will let you compute, say, the moebius function $\mu$ of your favorite poset given only the combinatorial data. In fact, it's so effective that you can see here for a sage package that will automatically compute these things for you!
If you want to read more about these, the original paper where the theory was developed is quite legible. See On the foundations of combinatorial theory I. Theory of Möbius Functions by Gian-Carlo Rota. For a textbook account, you can find this treated in Chapter 25 of van Lint and Wilson's A Course in Combinatorics, which is honestly a book worth having in its own right.
I hope this helps ^_^
