Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.

I explain myself better: if I take $X=\mathbb{P}^n$ and then I consider a subscheme $Y \subset X$, then to $Y$ corresponds an ideal (saturated if you want) $$I_Y=(F_1,\dots,F_r) \subset \mathbb{C}[x_0,\dots,x_n]$$ Sheafifying I can view $\mathcal{I}_Y$ as a coherent $\mathcal{O}_X-$module. I can also consider its twist by $\mathcal{O}_X(d)$, i.e. $$\mathcal{I} \otimes \mathcal{O}_X(d)=\mathcal{I}(d)$$ I have a very clear geometric meaning of sections of $\mathcal{I}(d)$, i.e. $H^0(\mathcal{I}(d))$ corresponds to hypersurfaces $F \in \mathbb{P}^n$ of degree $d$ containing $Y$. Now, if for example I take the Tangent sheaf $\mathcal{T}_X$, it is also a coherent $\mathcal{O}_X-$module, and geometrically $H^0(\mathcal{T}_X)$ is the space of vector fields over $\mathbb{P}^n$. Now, if I consider the twist $$\mathcal{T}_X \otimes \mathcal{O}_X(d)=\mathcal{T}_X(d)$$ what is the geometric meaning of the sections in $H^0(\mathcal{T}_X(d))$? Are they a sort of vector fields with special properties?

And more generally if I have a nice geometric description of sections of a coherent sheaf $H^0(\mathcal{F})$, how can I find a geometric interpretation of the sections of $H^0(\mathcal{F}(d))$? Thanks in advance for the help