The proof for (A ∪ B)' = (A' ∩ B') is:
Let's say x ∈ (A ∪ B)'. This means x ∉ (A ∪ B), x ∉ A and x ∉ B. So, x ∈ A' and x ∈ B'. So x ∈ (A' ∩ B')
So (A ∪ B)' ⊆ (A' ∩ B'). Then we have to prove (A' ∩ B') ⊆ (A ∪ B)' and we are done.
Can someone please explain why we write (A ∪ B)' ⊆ (A' ∩ B') and not (A ∪ B)' = (A' ∩ B')?
You have shown that $x\in (A\cup B)'\implies x\in (A'\cap B')$, in other words every element of the first set is contained in the second set, so the first set is contained into the second one, or $(A\cup B)'\subseteq(A'\cap B')$.
At this point you don't know that the $2$ sets are equal, there could be elements in the second set that are not contained in the first one, to rule out this possibility you have to prove $(A'\cap B')\subseteq(A\cup B)'$.
It is a very common strategy to prove $A\subseteq B$ and $B\subseteq A$ to show $B=A$ since it's usually easier to prove the $2$ statements separately then the equality directly