$\mathcal {B}$ is an exact abelian subcategory of an abelian category $\mathcal {A}$ such that $\mathcal {B}$ is closed in $\mathcal {A}$ under subobjects and quotient objects. I am not able to grasp what the author wants to say by this closed property?
Does it mean that any quotient of objects in $\mathcal {B}$ is in $\mathcal {A}$? I am a little bit confused.
I was trying to see via example of category of R/I modules $\subset$ category of R modules. But again does the closed property actually imply that every R/I module is an R module and any quotient of R/I module is also an R module.
Suppose $\mathbf{O}$ is an object in $\mathcal{B}$. Then it is also an object in $\mathcal{A}$. What "closed under subobojects" means is that if $\mathbf{O'}$ is a subobject of $\mathbf{O}$ in $\mathcal{A}$, then $\mathbf{O'}$ is also an object in $\mathcal{B}$, and it is also a subobject of $\mathbf{O}$ in $\mathcal{B}$.
Similarly, "closed under quotients": if $\mathbf{Q}$ is a quotient of $\mathbf{O}$ in $\mathcal{A}$, then $\mathbf{Q}$ is also an object in $\mathcal{B}$, where it is also a quotient of $\mathbf{O}$.
For your example: let $M$ be an $R/I$ module. Then viewing it as an $R$-module, it has $R$-submodules and $R$-quotient modules. But every $R$-submodule $N$ of $M$ is also an $R/I$-submodule of $M$; and every $R$-quotient module $Q$ of $M$ is also an $R/I$-quotient module of $M$.