In algebraic geometry by Robin Hartshorne, exercise II.6.3.a is written as follows
Cones. In this exercise, we compare the class group of a projective variety $V$ to the class group of its cone (I, Ex. 2.10). So let $V$ be a projective variety in $\mathbf{P}^n$, which is of dimension $\geqslant 1$ and nonsingular in codimension 1. Let $X=C(V)$ be the affine cone over $V$ in $\mathbf{A}^{n+1}$, and let $\bar{X}$ be its projective closure in $\mathbf{P}^{n+1}$. Let $P \in X$ be the vertex of the cone.
(a) Let $\pi: \bar{X}-P \rightarrow V$ be the projection map. Show that $V$ can be covered by open subsets $U_i$ such that $\pi^{-1}\left(U_i\right) \cong U_i \times \mathbf{A}^1$ for each $i$, and then show as in (6.6) that $\pi^*: \mathrm{Cl} V \rightarrow \mathrm{Cl}(\bar{X}-P)$ is an isomorphism. Since $\mathrm{Cl} \bar{X} \cong$ $\mathrm{Cl}(\bar{X}-P)$, we have also $\mathrm{Cl} V \cong \mathrm{Cl} \bar{X}$.
I don't understand the definition of the projection map. I do understand the definition of the projection from $C(V) -\{0\}\to V$ but not as defined in the exercise
Fix coordinates so that $P$ is $[0:\cdots:0:1]$. Then the formula for the projection map $\Bbb P^{n+1}\setminus \{P\} \to\Bbb P^n$ is $[x_0:\cdots:x_n:x_{n+1}] \mapsto [x_0:\cdots:x_n]$. To make this a morphism of schemes, we may work on the affine charts $D(t_i)$ for $0\leq i \leq n$. The chart $D(t_i)$ in $\Bbb P^{n+1}$ is isomorphic to $\operatorname{Spec} k[t_0,\cdots,t_{i-1},t_{i+1},\cdots,t_{n+1}]$, while the chart $D(t_i)$ in $\Bbb P^n$ is isomorphic to $\operatorname{Spec} k[t_0,\cdots,t_{i-1},t_{i+1},\cdots,t_n]$, and the projection morphism is given by $\operatorname{Spec}$ of the inclusion of rings $k[t_0,\cdots,t_{i-1},t_{i+1},\cdots,t_n] \to k[t_0,\cdots,t_{i-1},t_{i+1},\cdots,t_{n+1}]$. It is not hard to see that these morphisms glue together.
Alternately, if you read section II.7, you can find some other tools to construct the projection morphism. For a ring $A$ and a scheme $X$ over $A$, $A$-morphisms $X\to \Bbb P^n_A$ correspond to line bundles $\mathcal{L}$ on $X$ equipped with a choice of global sections $s_0,\cdots,s_n\in\mathcal{L}(X)$ such that $s_0,\cdots,s_n$ generate $\mathcal{L}$. Applying this construction with $X=\Bbb P^{n+1}\setminus \{P\}$, $\mathcal{L}=\mathcal{O}_{\Bbb P^{n+1}}(1)|_{\Bbb P^{n+1}\setminus\{P\}}$, and the $n+1$ sections $t_0,\cdots,t_n$, one gets the projection map.