Let $X$ be a measure space and $T:X\to X$ a measure preserving transformation. The Poincare recurrence theorem states that for any $T$ and any $A\subset X$ with $\mu(A) > 0$ (we take $\mu(X)=1$) there exists a natural number $n\in\mathbb{N}$ such that $\mu(A\cap T^{-n}A)>0$.
How can we use this result to show that if $A\subset X$ and $\mu(A)>0$, then for almost every $x\in A$, there exists a natural number $n\in\mathbb{N}$ such that $T^nx\in A$?
According to Wikipedia, what you are asking is the actual statement of the Poincare recurrence theorem, which follows almost directly from your first statement. If $A'$ denotes the set of all $x\in A$ such that $x$ does not return to $A$ in the orbit of $T$, we want to show that the measure of $A'$, $\mu(A')=0$. Suppose not. Then, by Poincaré recurrence, for some $n\in\mathbb{N}$, $\mu(A'\cap T^{-n}A)>0$, which means $A'\cap T^{-n}A\neq\emptyset$. Finally, if $x\in A'\cap T^{-n}A'$, then $x\in A'$ and $T^nx\in A'\subset A$, contradiction.
To prove the first statement, consider the sets $A, T^{-1}A,\ldots, T^{-k}A,\ldots, T^{-\ell}A,\ldots$, all of which have the same measure $\mu(A)$. Due to additivity of measure: $\mu(\bigsqcup A_i)=\sum\mu(A_i)$, there exist $k<\ell$ such that $$0<\mu(T^{-k}A\cap T^{-\ell}A)=\mu(A\cap T^{-(\ell-k)}A),$$ as desired.