Two definitions of $\mathcal{O}_{\mathbb{P}^{n}}(l)$

For any sheaf of $\mathcal{O}_{\mathbb{P}^n}$ modules $\mathcal{F}$ on $\mathbb{P}^n$, let $\Gamma_*(\mathcal{F})= \bigoplus_{n\in\mathbb{Z}} \Gamma(\mathcal{F}\otimes\mathcal{O}(n))$. Now, let $\mathcal{F}$ be a quasicoherent sheaf of $\mathcal{O}_{\mathbb{P}^n}$ modules. Then there is a natural isomorphism $\beta:\widetilde{\Gamma_*(\mathcal{F})}\to\mathcal{F}$ (this is Hartshorne II.5.15). After identifying $\Gamma_*(\mathcal{O}(l))=K[X_0,\cdots,X_n](l)$, we'll apply this statement to show what you want.

Identifying $\Gamma_*(\mathcal{O}(l))$: recall that $\Gamma(\mathcal{O}(a))$ is the set of homogeneous polynomials of degree $a$ in $X_0,\cdots,X_n$. This means that $\Gamma_*(\mathcal{O}(l))=\bigoplus_{n\in\mathbb{Z}} (K[X_0,\cdots,X_n])_{(n+l)}$, which exactly means that $\Gamma_*(\mathcal{O}(l))=K[X_0,\cdots,X_n](l)$.

Honestly calculating the transition functions should be possible using the isomorphism $\Gamma(\widetilde{M(l)}|_{D(f)},D(f))\cong (M_f)_l$ where the first subscript is localization and the second is "take the $l^{th}$ graded piece", but I get confused by this too.