What do time complexity reductions prove?
$A \leq_p B$ denotes that you can reduce $A$ to $B$ in polynomial time. That is, you can always transform an instance of $A$ to an instance of $B$ and you do this in polynomial time. Now if you have a polynomial time algorithm for solving an instance of $B$, you also have a polynomial time algorithm for solving an instance of $A$, clearly.

To show that $C$ is $\mathsf{NP}$-complete, you need two conditions to hold. First, $C$ must be in $\mathsf{NP}$. Second, every problem in $\mathsf{NP}$ is reducible to $C$ in polynomial time.