This is example 1.8.38 of Positivity in Algebraic Geometry I. The statement is that (after some translation) $I_X(d)$ is globally generated, where $X\subset \mathbb{P}^r$ is is a variety of degree $d$ and dimension $n$. The explanation given there is that we consider a linear subspace $\Lambda$ of dimension $r-n-1$, and consider the cone of $X$ centred at $\Lambda$, denoted by $C_\Lambda(X)$. Then this is a hypersurface of degree $d$. If we varying $\Lambda$, the resulting hypersurface generating $I_X(d)$.
My question is the last step. $I_X(d)$ means the degree $d$ polynomial, which gives a hypesurface, vanishing (contaning) $X$. Why we can conclude it is generated by global section there? I do not think I fully understand generated by global section geometrically very well.