The result has clearly been misstated or miscopied. The proof is correct for the following result:
If event $A$ and $B$ are events such that $P(A)$ and $P(B)$ are neither $0$ nor $1$, and $A$ is subset of $B$, then $A$ and $B$ are dependent events.
The reason for looking at $P(A\cap B)-P(A)P(B)$ is that by definition, $A$ and $B$ are independent if and only if $P(A\cap B)=P(A)P(B)$, i.e., if and only if $P(A\cap B)-P(A)P(B)=0$. But the hypothesis that $A\subseteq B$ implies that $P(A)=P(A\cap B)$, so $P(A\cap B)-P(A)P(B)=P(A)-P(A)P(B)=P(A)\big(1-P(B)\big)\;,$
which is $0$ if and only if either $P(A)=0$ or $1-P(B)=0$, i.e., if and only if either $P(A)=0$ or $P(B)=1$. These possibilities are ruled out by the corrected version of the hypothesis, so it must be the case that $P(A\cap B)-P(A)P(B)\ne 0$, $P(A\cap B)\ne P(A)P(B)$, and hence by definition $A$ and $B$ are not independent (which of course means that they are dependent).
The answer by copper.hat shows why the stated version is wrong.