Venn diagrams are not a formal proof, nor a substitute for it, just an illustrative tool that can be useful as a guiding tool for your narrative/proof.
If writing a formal proof for this law, you will need to show
$$A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C) \;\;\; \text{and} \;\;\; (A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C)$$
and then use the fact that if $X \subseteq Y$ and $Y \subseteq X$, then $X = Y$.
If you don't need formality, then in the appropriate context it can be used, I suppose. And, up to your ability to produce said diagrams, you could use a Venn diagram of $n$ circles, depending on what you're proving, but it gets messy quick so I wouldn't recommend it for more than $3$ sets.
In short, it depends on the level of formality that is expected of you. There's no denying that Venn diagrams in contexts like these are super, super helpful in illustrating concepts, and can be taken as a sort of heuristic proof, but they are not a substitute for formal proofs.
I say this in light of the assumption that you are probably encountering this in a class of some sort like a number of questions here. Classes in set theory, generally, will expect formality, not Venn diagrams, for example. In research, publications, journals, etc., things are much, much murkier depending on the context.
$x\in A\cup B$ means $x$ belongs to at least one of the sets between $A$ and $B$, might belong to both.
So $x\in A\cup B\cup C$ means $x$ belongs to at least one of the sets among $A,B,C$.
$x\in A^c \cup B^c \cup C^c$ means $x$ belongs to at least one of the sets $A^c,B^c,C^c$, that is to say,
for at least one set $A,B,C$, $x$ does not belong to it. So $x$ can belong to at most two of them.
Edit: Here's an example as asked by OP in the comments, consider the universe of tossing $3$ coins, i.e the entire sample space is $\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$ and consider the events:
$A=$ exactly one head
$B=$ at least one tail
$C=$ last toss is a head
so that we have the corresponding sets:
$A=\{HTT,THT,TTH\}$
$B=\{HHT,HTH,THH,HTT,THT,TTH,TTT\}$
$A=\{HHH,HTH,THH,TTH\}$
so the event "at most two of $A,B,C$ occur" is the set of those combinations in the sample space where all three don't occur (check below)
$(A\cap B \cap C)^c= \{TTH\}^c=\{HHT,HTH,THH,HTT,THT,HHH,TTT\}$
Best Answer
Venn diagrams can be a bit awkward when dealing with power sets. You could look at a couple of small examples, but in this case your best approach may be simply to try to prove it, and either succeed or see where you run into difficulties; the latter often gives a clue towards finding a counterexample.
So try to show first that $\wp(A\cap B)\subseteq\wp(A)\cap\wp(B)$. Suppose that $X\in\wp(A\cap B)$; then $X\subseteq A\cap B$. Therefore ...
If you can complete that argument successfully, you can try showing the opposite inclusion.