As in the picture says, hartshorne says there exists a exact sequece of sheaves $$0 \to \mathcal{L}(D-P) \to \mathcal{L}(D) \to k(p) \to 0 $$ where $\mathcal{L}(D)$ means associated invertible sheaves with the corresponding cartier divisor D. I have no idea how does this exact sequence of invertible sheaves come from. Can anyone give some hints? Thanks!
By Proposition II.6.18, you have that $\mathcal{L}(-P)$ is the ideal sheaf of the closed subscheme $\iota:\{P\}\hookrightarrow X$ (with reduced induced closed subscheme structure). Hence we have a SES $$ 0\to\mathcal{L}(-P)\to\mathcal{O}_X\to\iota_*\mathcal{O}_{\{P\}}\to 0. $$ Now tensor with $\mathcal{L}(D)$ and use the projection formula on the last term to see that it stays the same, and convince yourself that $\iota_*\mathcal{O}_{\{P\}}=k(P)$.