Your summary does not contain all the power of the subset axiom schema. It is not only intersection, it is also union, difference, as well quantified formulas.
To be more accurate for any formula in the language of set theory if we fix parameters so the formula has only one free variable $x$, then the collection of all those which are members of a particular set and satisfy this formula is a set.
Formally this means that if $\varphi(x,y_1,\ldots,y_k)$ is a formula in the language of set theory we have the following axiom: $$\forall p_1\ldots\forall p_k\forall A\exists B\bigg(u\in B\leftrightarrow\Big(u\in A\land\varphi(u,p_1,\ldots,p_k)\Big)\bigg)$$
which says that once we fix the parameters then for every set $A$ there is a set $B$ which is exactly the subset of $A$ consisting of those satisfying $\varphi$ (with the chosen parameters).
The importance of this axiom schema is to allow us and generate new sets. This is not only the union and the intersection of sets. The formulas can get increasing complicated which will then allow us to generate more and more sets. The parameters can vary greatly and increase the complexity.
For example, one example would be $\varphi(x,p_1)$ holds if and only if $p_1$ is $\omega+1$ (the natural numbers+a point at infinity) and $x$ is a set such that there exists $R\subseteq x\times x$ and $f\subseteq x\times p_1$ a bijection such that $f$ witnesses an order isomorphism between $(x,R)$ and $(p_1,\in)$.
If we apply this formula on $A=\mathcal P(\mathbb R)$ we will have all those sets of real numbers which are well-ordered with order type $\omega+1$. This is not "merely" an intersection of two sets. This is something I may use in an actual proof.
The last line, too, should hint that this is useful for proofs. The universe is big and contains a lot of things we don't know about in details. However it also known not to contain things we which do know about in details. The axiom schema of subset (or the stronger one, replacement) allow us to ensure the existence of sets when we need them (e.g. during proofs).