From Teschl's ODE book:
If one solution $\phi_1(t)$ is known [to the system $\frac{d}{dt}\vec{y}(t) = A(t)\vec{y}(t)$], one can use the following method known as a reduction of order: At least one component of $\phi_1(t)$ is nonzero, say the first one for notational simplicity. Let $X(t)$ be the identity matrix with the first column replaced by $\phi_1(t)$...
Now, I'm a little confused by what he means by "at least one component of $\phi_1(t)$ is nonzero". I believe he's asserting this so that $X(t)$ can be invertible at each value of $t$. I agree that for each $t$, at least one component of $\phi_1(t)$ should be nonzero (or else $\phi_1$ is just the zero solution for all time). That is, $(\forall t)\,(\exists n)\,(\phi_{1,n}(t)\neq 0).$ But in order that $X(t)$ as defined remain invertible at every point, it is necessary that $\phi_{1,1}(t)$ remain nonzero at each point. Why should this be true? Why should it be true that $(\exists n)\,(\forall t)\,(\phi_{1,n}(t)\neq 0)$?
Reduction of order is a practical matter of getting a formula for the solution. Once you have a formula, you can check that it works without justifying any steps you took to find the formula.
What it means in this context: we pick $t_0$. The vector $\phi_1(t_0)$ is not the zero vector. We pick a nonzero component; imagine it's the first one. This component remains nonzero in some neighborhood of $t_0$. Subsequent computations are valid in that neighborhood. At the end of computation we have a formula for the solution. So far it's been proved to work only in that neighborhood, but we can check it works everywhere (as long as solution itself is defined). One can refer to a general theorem to that effect: when coefficients are real-analytic functions, so is the solution, and a real-analytic function is determined by its values on any open interval. But "plug and see that it works" is easier.