You can undertstand this by thinking about the simplest case $n=2$, the case $n \geq 3$ being analogous.

The blow-up of $\mathbb{P}^2$ at a point $P$ is the Hirzebruch surface $\mathbb{F}_1$, and the two projections $$p_1 \colon \mathbb{F}_1 \longrightarrow \mathbb{P}^2, \quad p_2 \colon \mathbb{F}_1 \longrightarrow \mathbb{P}^1$$ are the blow-up morphism and the $\mathbb{P}^1$-fibration induced by the pencils of lines through $P$, respectively. So, denoting by $f$ the fibre of $p_2$, we have $$p_1^*\, \mathcal{O}(1) = E+f, \quad p_2^* \,\mathcal{O}(1) = f,$$ that is $p_2^*\,\mathcal{O}(1)=p_1^*\,\mathcal{O}(1)\otimes \mathcal{O}(-E).$

You can undertstand this by thinking about the simplest case $n=2$, the case $n \geq 3$ being analogous.

The blow-up of $\mathbb{P}^2$ at a point $P$ is the Hirzebruch surface $\mathbb{F}_1$, and the two projections $$p_1 \colon \mathbb{F}_1 \longrightarrow \mathbb{P}^2, \quad p_2 \colon \mathbb{F}_1 \longrightarrow \mathbb{P}^1$$ are the blow-up morphism and the $\mathbb{P}^1$-fibration induced by the pencil of lines through $P$, respectively. So, denoting by $f$ the fibre of $p_2$, we have $$p_1^*\, \mathcal{O}(1) = E+f, \quad p_2^* \,\mathcal{O}(1) = f,$$ that is $p_2^*\,\mathcal{O}(1)=p_1^*\,\mathcal{O}(1)\otimes \mathcal{O}(-E).$