Suppose $A$ and $B$ are disjoint sets in $X$ with complements $A^c$, $B^c$.
You know that each $x \in X$ that isn't in $B$ is in $B^c$. You also know that $a \in A \Rightarrow a \not\in B$ because $A$ and $B$ are disjoint. This implies that $A \subseteq B^c$.
Because $A \cup A^c = X$, $B^c \cup A^c = X $.
For simple set theory, the visualization of sets as groups of points on a two dimensional plot is intuitive.
If you color in $A$ and $B$, the part of $U$ that isn't colored is the subset of $U$ not in $A$ or $B$. If you color in the complements $A^c$ and $B^c$, then $B^c$ overlaps $A$, the part of the graph which isn't covered by $A^c$. That is essentially a hand-wavy version of the proof I wrote above.
Obviously these kinds of representations are not acceptable proofs, but they can help build intuition and tie together your ability to write proofs with visualization of mathematical properties.