If $Y\subseteq\mathbb{A}_n$ is an affine variety, we identify $\mathbb{A}^n$ with an open set $U_0\subseteq\mathbb{P}^n$ by the homeomorphism $\varphi_0\colon U_0\to\mathbb{P}^n$ defined by $\varphi_0([a_0:\dots: a_n])=(a_1,\dots, a_n)$. Then we can speak of $\overline{Y}$, the closure of $Y$ in $\mathbb{P}^n$, which is called the projective closure of $Y$.
Question Could someone explain to me the meaning of this thing? Why can we talk about closure in $\mathbb{P}^n$ only after this identification?