Definition of Closed Under Left Polynomial Composition
The correct interpretation is as follows. A class $\mathcal{C}$ is closed under left polynomial composition if the following holds:

For any $f(n) \in \mathcal{C}$ and any polynomial $p(n)$ there exists $g(n) \in \mathcal{C}$ such that $p(f(n)) = O(g(n))$.

In particular, the class $O(n)$ isn't closed under left polynomial composition.
Why are we interested in this definition? Here is an example. We can convert a $k$-tape Turing machine running in time $T(n)$ into a single-tape Turing machine running in time $5kT(n)$, see this question. If we start with a multi-tape Turing machine running in linear time, we don't necessarily end up with a single-tape Turing machine running in linear time. But if we start with a multi-tape Turing machine running in polynomial time, then we do end up with a single-tape Turing machine running in linear time. It thus makes sense that $O(n)$ isn't closed under left polynomial composition, but $n^{O(1)}$ is.